Algebraic Weaves and Braid Varieties
[2010]Primary: 53D10. Secondary: 53D15, 57R17.

R. Casals]Roger Casals
Dept. of Mathematics
University of California, Davis
One Shields Avenue, Davis, CA 95616
E. Gorsky]Eugene Gorsky
Dept. of Mathematics
University of California, Davis
One Shields Avenue, Davis, CA 95616
M. Gorsky]Mikhail Gorsky
Hausdorff Research Institute for Mathematics, Poppelsdorfer Allee 45, 53115 Bonn, Germany
J. Simental]José Simental
Dept. of Mathematics
University of California, Davis
One Shields Avenue, Davis, CA 95616

In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and  use these correspondences to obtain interesting stratifications of braid varieties and their quotients. It is shown that the maximal charts of these stratifications are exponential Darboux charts for the holomorphic symplectic structures, and we relate these strata to exact Lagrangian fillings of Legendrian links.

§ INTRODUCTION

This article studies braid varieties, a class of affine algebraic varieties associated to positive braids, and their relation to contact and symplectic geometry. First, the geometric properties of braid varieties are studied, including the construction of torus actions and holomorphic symplectic structures on their quotients. Then, we construct correspondences between these braid varieties by using certain moduli spaces associated to weaves, a class of labeled planar diagrams. These geometric correspondences are shown to induce valuable stratifications for braid varieties and their quotients, also unifying known constructions of A. Mellit, in the case of character varieties, and M. Henry and D. Rutherford, in the case of augmentation varieties.

The diagrammatic calculus based on weaves, presented in Section <ref>, allows for direct and explicit computations, and we provide new constructions of embedded exact Lagrangian fillings for Legendrian links through combinatorial methods. The main results of the article are Theorems <ref>, <ref>, <ref> and <ref>, and several detailed examples are provided throughout the manuscript. In particular, we believe that both the construction of a holomorphic symplectic structure on augmentation varieties, developed in Section <ref>, and the relation between weaves and cluster structures, as discussed in Section <ref>, is of value for contact and symplectic geometry.

§.§ Context
Legendrian links in contact 3–manifolds [1, 4, 38] are central in contact and symplectic geometry. Legendrian fronts, immersed planar cuspidal curves, arise in topology, as Cerf diagrams [2, 10, 24], in differential equations, as Stokes data for irregular singularities [5, 87, 88], and in analysis, as wavefront sets [49, 50, 61]. In this article, we use that a positive braid $\beta$ naturally gives rise to a Legendrian link $\La(\beta)\sse(\R^3,\xi_\st)$ [16, 38].

Associated to a Legendrian link $\La\sse(\R^3,\xi_\st)$, there exist two geometrically defined moduli spaces: the moduli space of microlocal sheaves in $\R^2$ microlocally supported at $\La$ [44, 56, 57], and the moduli space of exact Lagrangian fillings $L\sse(\R^4,\omega_\st)$, with boundary $\dd L=\La$ [2, 38, 16]. Note that the latter can be understood as the (geometric part of the) moduli space of objects of the Fukaya category of $(\R^4,\omega_\st)$ partially wrapped at $\La$. For the Legendrian links $\La(\beta)\sse(\R^3,\xi_\st)$, these moduli are algebraic stacks, often smooth algebraic varieties. The present manuscript studies a collection of algebraic varieties associated to a positive braid $\beta$, including and generalizing these two moduli spaces, and new correspondences between them. These algebraic correspondences are often induced by geometric exact Lagrangian cobordisms between Legendrian links, and can in general be described with a diagrammatic calculus, as we will show, building on the recent work of the first author with E. Zaslow [19].

In summary: we introduce the class of braid varieties, study torus actions and their quotients, construct correspondences and morphisms between them, and develop a diagrammatic calculus associated to these correspondences. As we establish these results, we prove several theorems of interest, including the fact that the augmentation variety associated to $\La(\beta)$ admits a holomorphic symplectic structure, and explain the relation between A. Mellit's stratification of character varieties [68] and the ruling stratification of the augmentation variety [45, 46]. Note that holomorphic symplectic structures play a central role in the study of moduli spaces of connections [7, 8], and there ought to be a relation to their symplectic structures through understanding the moduli stack of objects in the $\mbox{Aug}_+$-category [72] as a wild character variety [6, 86]. It should be noted that our diagrammatic calculus, which we refer to as algebraic weaves, provides a combinatorial and explicit approach to these stratifications. In addition, the strata are compatible with the holomorphic symplectic structure, the open toric charts admitting (exponential) holomorphic Darboux coordinates.

Finally, the manuscript illustrates several new connections to braid varieties and algebraic weaves that remain to be explored. For instance, a relation to cluster algebras is partially detailed at the end of the article. We believe that the calculus of algebraic weaves, as described in the present manuscript, may prove useful in the study of cluster algebras, character varieties (including irregular singularities), subword complexes and Rouquier complexes, see Section <ref>.

§.§ Main Results
Let $\gamma$ be a positive $n$-braid word $[\gamma]\in \Br^+_n$, $\gamma=\sigma_{i_1}\cdots \sigma_{i_\ell}$, and $\pi\in \GL(n,\C)$ a permutation matrix. Associated to these data, we consider the braid variety

X_0(\gamma;\pi):=\left\{(z_1,\ldots,z_\ell)\ :\ B_\gamma(z_1,\ldots,z_{\ell})\pi\ \text{is upper-triangular}\right\}\sse \C^{\ell},

where the matrix $B_{\gamma}(z_1,\ldots,z_{\ell})\in \GL(n,\C[z_1,\ldots,z_\ell])$ is defined to be the matrix product

B_{\gamma}(z_1,\ldots,z_{\ell}):=B_{i_1}(z_1)\cdots B_{i_\ell}(z_{\ell}),

and the matrices $B_i(z)\in \GL(n,\C[z])$ are defined by:

\[
(B_i(z))_{jk} := \begin{cases} 1 & j=k \text{ and } j\neq i,i+1 \\
1 & (j,k) = (i,i+1) \text{ or } (i+1,i) \\
z & j=k=i+1 \\
0 & \text{otherwise;}
\end{cases},\qquad\mbox{i.e.}\quad
1 & \cdots  & & & \cdots & 0\\
\vdots & \ddots & & & & \vdots\\
0 & \cdots & 0 & 1 & \cdots & 0\\
0 & \cdots & 1 & z & \cdots & 0\\
\vdots &  & & &\ddots & \vdots\\
0 & \cdots & & & \cdots & 1\\
\end{matrix}\right).
\]

These matrices $B_i(z)$ are referred to as braid matrices, and the only the non-trivial $(2\times 2)$-block is at $i$th and $(i+1)$st rows. Braid matrices have appeared in a range of areas, starting with L. Euler's continuants [33], G. Stokes' study of irregular singularities [88] (see P. Boalch's [8]), M. Broué and J. Michel's work on Deligne-Lusztig varieties [11], P. Deligne's braid invariants [25], and more recently in T. Kálmán's study of the Legendrian Contact DGA [54] (see also [20]) and A. Mellit's results on the curious Lefschetz property for character varieties [68], among others.

From our definition above, it is simple to see that $X_0(\gamma;\pi)$ is isomorphic to $X_0(\gamma';\pi)$ if $[\gamma]=[\gamma']\in \Br_n$, i.e. if two positive words $\gamma,\gamma'$ represent the same $n$-braid, the resulting braid varieties are isomorphic, hence the name. In the course of the article, the permutation (matrix) $\pi$ will often be the identity $\pi=\mathrm{Id}=e\in S_n$ or $\pi=w_0=(n\ n-1\ \ldots\ 1)\in S_n$. Let $\Delta\in \Br^+_n$ be a positive braid lift of the permutation $w_0$, i.e. $\Delta$ will be a braid word for the half-twist.[See Example <ref> for our specific choice of positive braid word for $\Delta$.]

The first result of the article establishes geometric properties of braid varieties, including the existence of a torus action and their relation to the Floer-theoretically defined augmentation varieties [9, 23, 72]. It reads as follows:

Let $\gamma$ be a positive $n$-braid word $[\gamma]\in \Br^+_n$. Then the following statements hold:

(i) $X_0(\gamma\Delta;1)\simeq X_0(\gamma;w_0)\times \C^{\binom{n}{2}}$, and $X_0(\gamma;w_0)$ is non-empty if and only if the Demazure product of $\gamma$ equals $w_0$. In this case, $X_0(\gamma;w_0)$ is an irreducible complete intersection of dimension $\ell(\gamma)-\binom{n}{2}$, and $X_0(\gamma\Delta;1)$ is an irreducible complete intersection of dimension $\ell(\gamma)$.

Suppose that there exists a positive $n$-braid word $\beta$ such that $\gamma=\beta\Delta$. Then:

(ii) The braid variety $X_0(\beta\Delta;w_0)$, and thus $X_0(\beta\Delta^2;1)$, is smooth.

(iii) There exists a free torus $T$-action on $X_0(\beta\Delta;w_0)$ such that the quotient algebraic variety $X_0(\beta\Delta;w_0)/T$ is smooth and holomorphic symplectic.

(iv) There exists an isomorphism between $X_0(\beta\Delta;w_0)/T$ and an augmentation variety $\Aug(\beta)$ associated to the Legendrian link $\La(\beta)$. In particular, $\Aug(\beta)$ is a holomorphic symplectic $($smooth$)$ affine variety.[See Section <ref> for the details on marked points.]

In addition, the open Bott-Samelson variety $\OBS(\beta)$ associated to $\beta$ is isomorphic to the quotient

$$\OBS(\beta)\cong\left(\GL(n,\C)\times X_0(\beta;1)\right)/\SB,$$

where $\SB\sse \GL(n,\C)$ is the Borel subgroup of upper-triangular matrices. $\Box$

In Theorem <ref>.(iii), the dimension of the torus $T$ does depend on the number of components in the closure of $\beta$, see Section <ref> for details. The different varieties and the torus action featured in Theorem <ref> are presented in the course of the article, and the proof of this theorem is obtained by gathering some the results we develop, such as Theorem <ref>, Theorem <ref>, Theorem <ref> and Corollary <ref>.  See also Section <ref> for the definition of Demazure product, and note that the Demazure product of $\beta\Delta$ equals $w_0$ for any $\beta$.

In the development of this article, we learned that the authors of [37] were also studying holomorphic symplectic structures on augmentation varieties, working on a different construction with plabic graphs. Their work is yet to appear but, once it does, it would be interesting to compare these symplectic structures, as the constructions seem to be significantly different and a potential link between weaves and plabic graphs could be fruitful.$\Box$

Theorem <ref> discusses the absolute aspects of braid varieties. The study of such varieties also relies crucially on their relative geometry: morphisms between different such braids varieties and, more generally, correspondences, yield interesting (and useful) results. In order to study this relative setting, we develop the diagrammatic calculus of algebraic weaves, which we summarize as follows.

Let $\mathfrak{W}_n$ be the category defined as:

- Objects: $\mbox{Ob}(\mathfrak{W}_n)$ are arbitrary positive braid words $\gamma=\sigma_{i_1}\cdots \sigma_{i_\ell}$, $[\gamma]\in Br^+_n$,

- Morphisms: $\mbox{Hom}_{\mathfrak{W}_n}(\gamma,\gamma')$ are compositions of the four elementary moves

$$\sigma_i\sigma_{i}\rightarrow \sigma_i,\qquad \sigma_i\sigma_{i+1}\sigma_i\leftrightarrow \sigma_{i+1}\sigma_i\sigma_{i+1},\qquad\sigma_i\sigma_j\rightarrow \sigma_{j}\sigma_i\ (|i-j|>1),\quad\mbox{ and }\quad\sigma_i\sigma_i\leftrightarrow 1,$$

modulo certain relations, explicitly drawn in Section <ref>.

The morphisms in $\mathfrak{W}_n$ will be represented diagrammatically as certain planar graphs with edges decorated by simple transpositions $s_i$. (Namely, $s_i$ are the Coxeter projections of the Artin braid generators $\sigma_i$, $1\leq i\leq n$.) These planar graphs are referred to as weaves, following the notation in <cit.>, and $\mathfrak{W}_n$ will be called the category of algebraic weaves. The elementary moves above, i.e. the building blocks for morphisms, can be drawn as follows:

There is also a dual 6-valent vertex corresponding to $\sigma_i\sigma_{i-1}\sigma_i\to \sigma_{i-1}\sigma_i\sigma_{i-1}$ which we do not draw here.
An algebraic weave, obtained by (vertically) concatenating the models above, represents a morphism from the braid word on the top to the braid word on the bottom. The composition of weaves

is given by vertical stacking of these weave diagrams. See Figure <ref> for an instance of a morphism.

An algebraic weave in $\mbox{Hom}_{\mathfrak{W}_n}(\gamma,\gamma')$ between the two positive $3$-braids $\gamma=\sigma_1^3\sigma_2\sigma_1^3\sigma_1^2\sigma_1^3\sigma_2^3\sigma_1^3\sigma_2\sigma_1^3$, on the top, and $\gamma'=\sigma_2^3\sigma_1^2\sigma_2^2$, on the bottom. The color code is that  blue  is labeled with the transposition $s_1$ and  red  is labeled with $s_2$.

Note that this diagrammatic category is in part similar to the categories appearing in Soergel calculus [30, 31], but differs in several key aspects. In particular, in the category of algebraic weaves there is no requirement that the two ways of getting from $\sigma_i\sigma_i\sigma_i$ to $\sigma_i$, via the moves $\sigma_i\sigma_i\rightarrow\sigma_i$, are equivalent:

The difference between these diagrams will be referred to as a weave mutation.$\Box$

Let $\mathfrak{C}$ be the category of algebraic varieties whose morphisms are correspondences [64]. The second result in this manuscript shows that the braid varieties, and their correspondences, provide a realization of the weave category $\mathfrak{W}_n$:

There exists a functor $\mathfrak{X}_0:\mathfrak{W}_n\lr\mathfrak{C}$ such that:

(a) Objects: For a positive braid word $\gamma\in\mbox{Ob}(\mathfrak{W}_n)$, the functor $\mathfrak{X}_0$ associates the braid variety $\mathfrak{X}_0(\gamma):=X_0(\beta;w_0)$.

(b) Morphism: For a weave $\Sigma\in \Hom_{\mathfrak{W}_n}(\beta_0,\beta_1)$, the functor associates a correspondence $\mathfrak{X}_0(\Sigma)$ between
$X_0(\beta_0;w_0)$ and $X_0(\beta_1;w_0)$, such that correspondences $\mathfrak{X}_0(\Sigma)$ and $\mathfrak{X}_0(\Sigma')$ associated to equivalent weaves $\Sigma,\Sigma'$ are isomorphic. $($The algebraic variety $\mathfrak{X}_0(\Sigma)$ is in fact described as a certain moduli space governed by the weave $\Sigma$.$)$

(c) Composition: Let $\Sigma_1\in\Hom_{\mathfrak{W}_n}(\beta_0,\beta_1)$, $\Sigma_2\in\Hom_{\mathfrak{W}_n}(\beta_1,\beta_2)$, and their composition $\Sigma\in \Hom_{\mathfrak{W}_n}(\beta_0,\beta_2)$, which is obtained by concatenation of $\Sigma_1$ and $\Sigma_2$. Then the composition of weaves corresponds to the diagram:

𝔛_0(Σ) dl dr

𝔛_0(Σ_1) dl dr          𝔛_0(Σ_2) dl dr

X_0(β_0;w_0)          X_0(β_1;w_0)         X_0(β_2;w_0),

where the middle square is Cartesian.

(d) Let $\Sigma\in\Hom_{\mathfrak{W}_n}(\beta_0,\beta_1)$ be a weave with no caps, $a$ cups and $b$ trivalent vertices. Then the correspondence $\mathfrak{X}_0(\Sigma)$ defines an injective map

\mathfrak{X}_0(\Sigma):\C^a\times (\C^*)^b\times X_0(\beta_1;w_0)\hookrightarrow X_0(\beta_0;w_0).

Furthermore, the correspondences $\mathfrak{X}_0(\Sigma)$ are equivariant with respect to appropriate torus $T$-actions and, using Theorem <ref>.$($iv$)$, yield correspondences between augmentations varieties.

The proof of Theorem <ref> occupies the majority of Section <ref>, the equivariance statement being discussed in Subsection <ref>.

The statements in Theorem <ref>.(a)-(c) are the algebraic analogues of the symplectic geometric results obtained by the first author in [19]. In short, [19] shows that any weave $\mbox{Hom}_{\mathfrak{W}_n}(\beta_0,\beta_1)$ yields an exact Lagrangian cobordism $L(\Sigma)\sse(\R^4,\la_\st)$ from the Legendrian link $\La(\beta_1)\sse(\R^3,\xi_\st)$ to $\La(\beta_0)$. Equivalent weaves give rise to Hamiltonian isotopic Lagrangians, and certain weaves yield embedded exact Lagrangian cobordisms.$\Box$

It should be noted that Theorem <ref>.(d) provides a unifying framework for many known stratifications, including the ruling stratification in augmentation varieties [36, 45, 46], and the stratification by walks in character varieties [68]. In particular, note that the algebraic variety $X_0(\Delta;w_0)$ is a point, and thus Theorem <ref> implies the following:

Let $\Sigma\in\Hom_{\mathfrak{W}_n}(\gamma,\Delta)$ be a weave with no caps, $a$ cups and $b$ trivalent vertices, $a,b\in\N$. Then the correspondence $\mathfrak{X}_0(\Sigma)$ yields an injective map

$$\mathfrak{X}_0(\Sigma):\C^a\times (\C^*)^b\hookrightarrow  X_0(\gamma;w_0),\ 2a+b=\ell(\gamma)-\binom{n}{2}.$$

In fact, Theorem <ref> will prove that weaves can be used to provide many stratifications of the braid varieties $X_0(\gamma;w_0)$, with strata of the form
$\C^a\times (\C^*)^b$. In particular, weaves $\Sigma\in\mbox{Hom}_{\mathfrak{W}_n}(\gamma,\Delta)$ with no caps or cups yield (algebraic) toric charts $(\C^*)^{\ell(\gamma)-\binom{n}{2}}\subset X_0(\gamma;w_0)$ of maximal dimension, whose complement can also be stratified with weaves. These weaves, with no caps or cups, are referred to as Demazure weaves.

The manuscript also includes a new construction of weaves, coming from a class of labeled triangulations. This construction uses Demazure products in a crucial manner and provides a systematic (and combinatorial) mechanism to construct embedded exact Lagrangian fillings for Legendrian links $\La(\beta)\sse(\R^3,\xi_\st)$ which are obtained as the closure of a positive braid $\beta$.$\Box$

Finally, complementing Theorem <ref> and Theorem <ref>, we give a geometric interpretation to these toric charts associated to Demazure weaves $\Sigma\in\mbox{Hom}_{\mathfrak{W}_n}(\beta\Delta,\Delta)$, as follows.

First, we show in Section <ref> that these charts can be combinatorially obtained by opening the crossings of the positive braid $\beta$. Indeed, Section <ref> shows that there is an injective map

X_0(\beta'\Delta; w_0)\times \C^*\hookrightarrow X_0(\beta\Delta; w_0),

if the positive braid word $\beta'$ is obtained from $\beta$ by removing exactly one crossing. Therefore, opening the crossings in $\beta$ one by one, in some order, yields a toric chart in $X_0(\beta\Delta; w_0)$. Different orders might yield identical or different toric charts. For instance, for a 2-strand braid $\beta=\sigma_1^n$, there are $n!$ possible orderings and one obtains a Catalan number $C_n$ of toric charts. In particular, in this correspondence, each toric chart is obtained by exactly one 312-pattern avoiding permutation. Regarding this relation, between toric charts and openings of crossings, we establish the following result:

Let $[\beta]\in Br^+_n$ be a positive braid and $\beta=\sigma_{i_1}\cdot\sigma_{i_2}\cdot\ldots\cdot \sigma_{i_l}$ positive braid word. Consider an ordering $\rho\in S_{l}$ for the crossings of $\beta$. Then:

(i) There exists a $($Demazure$)$ weave $\Sigma_\rho$ such that the sequence of crossing openings according to $\rho$ is realized by the weave $\Sigma_\rho$. Conversely, any Demazure weave $\Sigma$ is equivalent to opening crossings for some ordering $\rho\in S_l$, i.e. there exists $\rho\in S_l$ such that $\Sigma$ is equivalent to $\Sigma_\rho$.

(ii) Two toric charts $C_1,C_2\sse X_0(\beta\Delta;w_0)$ associated to different orderings of the crossings are represented by weaves $\Sigma_1,\Sigma_2$ such that $\Sigma_1,\Sigma_2$ are related by a sequence of mutations. In addition, the union of all such toric charts covers $X_0(\beta\Delta;w_0)$ up to codimension 2.

The first item is proven in Lemma <ref> and Theorem <ref>, and the proof of the codimension-2 cover is established in Theorem <ref>. The mutation equivalence of any weaves yielding toric charts  follows from the more general Theorem <ref>, which states that, under technical conditions that are satisfied in the weaves pertaining to Theorem <ref>, any two Demazure weaves between the same two braid words are related by a sequence of equivalence moves and mutations. Note that Theorem <ref> is a translation of a result of B. Elias [29] to our weaves framework.

Note that both the openings of crossings and mutations can be described in terms of braid words. Indeed, consider a braid word $\sigma_iu\sigma_j$ with $\sigma_iu=u\sigma_j$, i.e. $(\sigma_i,\sigma_j)$ is deletion pair in the notation of [47]. Then, we can consider two different weaves:

σ_iuσ_j dl dr

σ_iσ_iu  d        uσ_jσ_jd

σ_iu rr        uσ_j

In this diagram, the left weave $\sigma_iu\sigma_i\to u\sigma_j$ corresponds to the opening of the crossing $\sigma_i$, and the right weave $\sigma_iu\sigma_i\to u\sigma_j$ to opening the crossing $\sigma_j$. Suppose that the positive braid word $u$ can be broken down in the middle $u=u_1\sigma_ku_2$. Then we obtain:

\sigma_iu\sigma_j=\sigma_iu_1\sigma_ku_2\sigma_j,\ \sigma_iu_1=u_1\sigma_k,\ \sigma_ku_2=u_2\sigma_j,

and the two associated weaves are either related by a sequence of mutations, or they are equivalent. Theorem <ref> ensures that any Demazure weave is equivalent to a weave composed from such pieces.$\Box$

We conclude by giving a topological interpretation to these toric charts. Given a weave $\Sigma$, we consider the associated spectral curve $L(\Sigma)$, constructed as a branched cover of the plane ramified along the trivalent vertices of the weave, following [19, 40, 86]. Then we show the following result:

Let $\beta$ be a positive braid word, and $\Sigma\in\mbox{Hom}_{\mathfrak{W}_n}(\beta\Delta,\Delta)$ a Demazure weave. Then, the following two statements hold:

(a) The toric chart $C(\Sigma)\sse\Aug(\beta)$ corresponding to the weave $\Sigma$ is naturally isomorphic to $\Spec(\Z[H_1(L(\Sigma),\Z)]) \times S$, where $S$ is a $2(k-1)$-dimensional symplectic torus that does not depend on $\Sigma$ and $k$ is the number of components of the closure of $\beta$.

(b) The restriction of the symplectic form to the toric chart $C(\Sigma)\cong\Spec(\Z[H_1(L(\Sigma),\Z)])$ corresponds to the intersection form on $H_1(L(\Sigma),\Z)$.$\Box$

The article [37] endows $X_0(\beta\Delta,w_0)$ and $\Aug(\beta)$ with the structure of cluster varieties, with toric charts being cluster charts, and mutations corresponding to cluster mutations. To be precise, [37] only works in characteristic 2, but can be lifted to characteristic 0 by using the first author's upcoming work [20]. We expect that the cluster coordinates correspond to some natural bases in $H_1(L(\Sigma))$. (The difference between cluster $X$-coordinates and cluster $A$-coordinates is yet to be understood in the context of augmentation varieties.) It would be interesting to describe these bases and cluster coordinates directly from a weave, see Section <ref> for a more detailed discussion.$\Box$

Acknowledgements: This paper started as a joint project with our colleague Monica Vazirani, and would not have happened without her; we thank her warmly. We also thank Ben Elias, Honghao Gao, Tamas Kálmán, Anton Mellit, Lenny Ng, Minh-Tam Trinh and Daping Weng for useful discussions. E. Gorsky would like to thank Yuri Chekanov for the lifetime influence.

R. Casals is supported by the NSF grant DMS-1841913, the NSF CAREER grant DMS-1942363 and the Alfred P. Sloan Foundation. The work of E. Gorsky was partially supported by the NSF grants DMS-1700814 and DMS-1760329. This work was completed during M. Gorsky's participation in the junior trimester program “New Trends in Representation Theory” at the Hausdorff Institute for Mathematics in Bonn. He is very grateful to the Institute for the perfect working conditions.$\Box$

§ BRAID VARIETIES AND AUGMENTATION VARIETIES

In this section we introduce and start studying braid varieties. Part of Theorem <ref> is proven in this section, with the holomorphic symplectic structure being discussed in Section <ref>. This section also discusses the torus actions on braid varieties and their quotients, which relate to augmentations varieties.

Notations for the braid group. Let $n\in\N$, the braid group $\Br_n$ on $n$-strands is presented with $n-1$ generators $\sigma_i$, $i\in[1,n-1]$, and relations

\begin{equation}
\label{eq:braid}
\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_{i}\sigma_{i+1},\qquad \sigma_i\sigma_j=\sigma_j\sigma_i\ \text{for}\ |i-j|\ge 2, i,j\in[1,n-1].
\end{equation}

In this article, we mainly work with the positive braid monoid $\Br^+_n\sse \Br_n$ generated by the nonnegative powers of the generators $\sigma_i$, $i\in[1,n-1]$. By definition, a (positive) braid word is a product expression of non-negative powers of the generators $\sigma_i$ where no relations are being applied. For instance, the two braid words $\sigma_1\sigma_2\sigma_3\sigma_1$ and $\sigma_2\sigma_1\sigma_2\sigma_3$ are distinct as braid words and represent the same element $[\sigma_1\sigma_2\sigma_3\sigma_1]=[\sigma_2\sigma_1\sigma_2\sigma_3]\in\Br^+_4$.

The symmetric group $S_n$ is the Coxeter group associated to $\Br_n$: it is generated by the transpositions $s_i=(i\ i+1)$, subject to relations (<ref>) above and the additional relation $s_i^2=1$, for all $i\in[1,n-1]$. By definition, a reduced expression for a permutation $w\in S_n$ is a minimal length expression for the element $w$ as a product of the generators $s_i$, $i\in[1,n-1]$; the length $\ell(w)$ is defined as the length of such reduced expression. It is well-known that any two reduced expressions are related by a sequence of braid moves (<ref>). Therefore, one can define a  positive braid lift of $w\in S_n$ to $\Br^+_n$ by choosing an arbitrary reduced expression and replacing each generator $s_i$ with the generator $\sigma_i$, for each $i\in[1,n-1]$. We will refer to such positive braid lifts as reduced braid words. To ease notation, we interchangeably use $\sigma_i, \, s_i,$ and sometimes simply $i$ for the braid group generators, $i\in[1,n-1]$.

§.§ Braid matrices and braid varieties

Braid varieties are affine algebraic varieties cut out by matrix equations. Their definition relies on the following notion:

Let $n\in\N$, $i\in[1,n-1]\in\N$ and $z$ a (complex) variable. The braid matrix $B_i(z)\in \GL(n,\C[z])$ is defined as

(B_i(z))_{jk} := \begin{cases} 1 & j=k \text{ and } j\neq i,i+1 \\
1 & (j,k) = (i,i+1) \text{ or } (i+1,i) \\
z & j=k=i+1 \\
0 & \text{otherwise;}
\end{cases},\qquad\mbox{i.e.}\quad
1 & \cdots  & & & \cdots & 0\\
\vdots & \ddots & & & & \vdots\\
0 & \cdots & 0 & 1 & \cdots & 0\\
0 & \cdots & 1 & z & \cdots & 0\\
\vdots &  & & &\ddots & \vdots\\
0 & \cdots & & & \cdots & 1\\
\end{matrix}\right).

Given a positive braid word $\beta=\sigma_{i_1}\cdots\sigma_{i_r} \in \mbox{Br}_{n}^{+}$  and $z_{1}, \dots, z_{r}$ complex variables, we define the braid matrix $B_{\beta}(z_1,\ldots,z_r)\in\GL(n,\C[z_1,\ldots,z_r])$ to be the product

B_{\beta}(z_1,\ldots,z_r)=B_{i_1}(z_1)\cdots B_{i_r}(z_r).

For instance, it follows from Definition <ref> that $B_{\beta}(0,\dots, 0)$ is simply the permutation matrix associated to the Coxeter projection $\pi(\beta)\in S_n$. Thus, in a sense, braid matrices are deformations of permutation matrices. It is a simple computation to verify the two relations:

\begin{equation}\label{equation: reidemeister3 matrices}
B_{i}(z_1)B_{i+1}(z_2)B_{i}(z_3)=B_{i+1}(z_3)B_{i}(z_2-z_1z_3)B_{i+1}(z_1),\quad \forall i\in[1,n-2],
\end{equation}

and $B_{i}(z_{1})B_{j}(z_{2}) = B_{j}(z_{2})B_{i}(z_{1})$, for $|i - j|\geq2$. Here is a first pair of useful examples:

Consider the positive braid word $\Delta = (\sigma_{1}\sigma_{2}\cdots\sigma_{n-1})(\sigma_{1}\cdots\sigma_{n-2})\cdots(\sigma_{1}\sigma_{2})\sigma_{1}$, which represents a half-twist. Its associated braid matrix is

B_{\Delta}\left(z_{1}, \dots, z_{\binom{n}{2}}\right) = \left(\begin{matrix} 0 & 0 & \ldots & 1\\ 0 & \ldots & 1 & z_{1} \\
\vdots & \cdots & \cdots & z_{n-2}\\
1 & z_{\binom{n}{2}} & \cdots & z_{n-1}\\
\end{matrix}\right).

Let $\Delta'\in \Br^+_n$ be any positive braid lift of the longest element $w_{0}$ of $S_{n}$. It then follows from the braid relation (<ref>) that

B_{\Delta'}\left(z_{1}, \dots, z_{\binom{n}{2}}\right) = \left(\begin{matrix} 0 & 0 & \ldots & 1\\ 0 & \ldots & 1 & z_{2,n} \\
\vdots & \cdots & \cdots & z_{n-1,n}\\
1 & z_{n,2} & \cdots & z_{nn}\\
\end{matrix}\right),

where the $z_{i,j}$ are algebraically independent polynomials in $\C[z_{1}, \dots, z_{\binom{n}{2}}]$.$\Box$

Let $\Delta^{2}\in \Br^+_n$ represent the full-twist braid, i.e. the square of the positive braid lift of $w_0\in S_n$ to the braid group. Then its braid matrix can be decomposed as

B_{\Delta^{2}}\left(z_{1}, \dots, z_{\binom{n}{2}}, w_{1}, \dots w_{\binom{n}{2}}\right) =LU =\left(\begin{matrix} 1 & 0 & \ldots & 0\\ c_{21} & 1 & \ldots & 0 \\
\vdots & \cdots & \ddots & 0\\
c_{n1} & \cdots & \cdots & 1\\
\end{matrix}\right)\left(\begin{matrix} 1 & u_{12} & \ldots & u_{1n}\\ 0 & 1 & \ldots & u_{2n} \\
0 & \cdots & \ddots & u_{n-1,n}\\
0 & \cdots & \cdots & 1\\
\end{matrix}\right),

where $c_{ij} \in \C[z_{1}, \dots, z_{\binom{n}{2}}]$ and $u_{ij} \in \C[w_{1}, \dots, w_{\binom{n}{2}}]$ are algebraically independent.

By Example <ref>, $B_{\Delta}= L w_0=w_0U$. Hence $B_{\Delta^{2}}=B_{\Delta}B_{\Delta}=Lw_0w_0U=LU$.

Let us now use braid matrices to define the central object of interest in this manuscript:

Let $\beta=\sigma_{i_1}\cdots\sigma_{i_r} \in \mbox{Br}_{n}^{+}$ be a positive braid word. The braid variety $X_{0}(\beta)\sse\C^r$ associated to $\beta$ is the affine closed subvariety given by

X_{0}(\beta):=\left\{(z_1,\ldots,z_r) : B_{\beta}(z_1,\ldots,z_r)\mbox{ is upper-triangular}\right\}\sse\C^{r}.

Let $\pi \in S_{n}$ be considered as a permutation matrix. We define the braid variety $X_{0}(\beta; \pi) \sse \C^{r}$ as

X_{0}(\beta; \pi) :=\left\{(z_1, \ldots, z_r) : B_{\beta}(z_1,\ldots,z_r)\pi \mbox{ is upper-triangular}\right\}\sse\C^{r}.$$

It follows from the braid relation (<ref>) that different presentations of the same braid $[\beta]\in \Br_n$ yield algebraically isomorphic braid varieties.$\Box$

Let us give some simple examples of braid varieties.

Consider the positive braid associated to the full twist $\beta = \FT$. Lemma <ref> implies that $X_{0}(\FT)$ is given by the equations $c_{ij} = 0$, and thus the braid variety is the affine space $X_{0}(\FT) \cong \C^{\binom{n}{2}}$. Similarly, Example <ref> implies that the braid variety $X_{0}(\Delta; w_{0}) = \{\operatorname{pt}\}$ is a point.$\Box$

In fact, the computation in Example <ref> shows that $X_{0}(\beta; w_{0})$ admits a closed embedding into $X_{0}(\beta\cdot\Delta)$, and the image is given by the equations $z_{ij} = 0$. In general, if $\Pi \in \mbox{Br}^{+}_{n}$ is a positive lift of a permutation $\pi \in S_{n}$ then $X_{0}(\beta; \pi)$ embeds into $X_{0}(\beta\cdot\Pi)$. Let us now establish the general dimension and smoothness for braid varieties:

Let $\beta\in \Br_n^+$ be a positive braid of length $\ell(\beta)$. Then, the braid varieties $X_{0}(\beta\cdot \Delta; w_0)$ and $X_{0}(\beta\cdot\FT)$ are smooth of dimension $\ell(\beta)$ and $\ell(\beta)+\binom{n}{2}$, respectively. In addition, $X_{0}(\beta\cdot\FT)\simeq X_{0}(\beta\cdot \Delta; w_0)\times \C^{\binom{n}{2}}$.

The variety $X_0(\beta\cdot \FT)$ is defined by the condition that $B_{\beta\cdot \FT}$ is an upper triangular matrix.
By Lemma <ref>, we get

B_{\beta\cdot \FT} = B_{\beta}B_{\FT}=B_{\beta}LU=B_{\beta \cdot \Delta}w_0U.

This is upper-triangular if and only if $B_{\beta \cdot \Delta}w_0$ is upper-triangular, which is precisely the condition defining $X_{0}(\beta\cdot \Delta; w_0)$. Therefore, $X_{0}(\beta\cdot \FT)\simeq X_{0}(\beta \cdot \Delta; w_0)\times \C^{\binom{n}{2}}$, with $\C^{\binom{n}{2}}$ being the coordinates on the upper uni-triangular matrix $U$. Next, note that $B_{\beta\cdot \Delta}w_{0} = B_{\beta}L$ and we can write $B_{\beta}L=U'$, then

Note that the existence of an $LU$ decomposition $M=LU''$ is an open condition on $M$, namely the non-vanishing of principal minors; also, if an $LU$ decomposition exists, it is unique provided that $L$ has 1s on the diagonal. Therefore $X_{0}(\beta\cdot \Delta; w_{0})$ is isomorphic to an open subset in the affine space $\C^{\ell(\beta)}$. Hence, it is smooth of dimension $\dim X_{0}(\beta\cdot\Delta; w_0) = \ell(\beta)$, and $X_0(\beta\cdot \FT)$ is also smooth of
dimension $\dim X_0(\beta\cdot \FT)=\ell(\beta)+\binom{n}{2}$, as required.

Note that a similar smoothness result was proved in <cit.>. The braid varieties associated to 2-stranded braids $\beta\in \Br^+_2$ are smooth varieties whose equations closely relate to Euler's continuants [33]. Let us consider the following instance:

$($Trefoil Knot$)$ Consider $\beta=\sigma_1^3\in Br^+_2$, whose $(-1)$-framed closure yields the $($max-tb$)$ right-handed trefoil knot. The braid variety $X_{0}(\sigma_1^5) = X_{0}(\sigma_{1}^{3} \cdot \Delta^2)$ is defined by the equation:

B(z_1)B(z_2)B(z_3)B(z_4)B(z_5)\mbox{ is upper-triangular}.

This condition can written as

B(z_1)B(z_2)B(z_3)\left(\begin{matrix} 1 & 0\\ z_4 & 1\end{matrix}\right)\left(\begin{matrix} 1 & z_5\\ 0 & 1\end{matrix}\right)\mbox{ is upper-triangular},

and equivalently

B(z_1)B(z_2)B(z_3)\left(\begin{matrix} 1 & 0\\ z_4 & 1\end{matrix}\right)=\left(\begin{matrix} z_2+(z_2z_3+1)z_4 & z_2z_3+1\\ z_1z_2+(z_1+(z_1z_2+1)z_3)z_4+1 & z_1+(z_1z_2+1)z_3\end{matrix}\right)\mbox{  upper-triangular}.

Note that we have

$$\left(\begin{matrix} 1 & 0\\ z_4 & 1\end{matrix}\right)=\left(\begin{matrix} 0 & 1\\ 1 & z_4\end{matrix}\right)\left(\begin{matrix} 0 & 1\\ 1 & 0\end{matrix}\right)=\left(\begin{matrix} 0 & 1\\ 1 & z_4\end{matrix}\right)w_0,$$

and thus the condition above, in the coordinates $(z_1,z_2,z_3,z_4)\in\C^4$, is in fact the equation for $X_{0}(\sigma_1^3\cdot \Delta; w_{0})$. This readily implies that $X_{0}(\sigma_1^{3}\cdot \FT)$ is isomorphic to $X_{0}(\sigma_1^3\cdot \Delta; w_{0})$ times an affine line $\C=\Spec(\C[z_5])$, as generally proven in Theorem <ref>. It thus suffices to understand $X_{0}(\sigma_1^3\cdot \Delta; w_{0})$. For that, consider the equation above:

\begin{equation}
\label{eq:char trefoil}
X_{0}(\sigma_1^3\cdot \Delta; w_{0})=\{(z_1,z_2,z_3,z_4)\in\C^4:1+z_1z_2+z_4(z_1+z_3+z_1z_2z_3)=0\}\sse\C^4,
\end{equation}

which cuts out a hypersurface, and should be smooth according to Theorem <ref>. Indeed, note that we must have $z_1+z_3+z_1z_2z_3\neq0$, otherwise the defining Equation <ref> would imply $1+z_1z_2=0$, and in these constraints $z_1=z_1+z_3(z_1z_2+1)=z_1+z_3+z_1z_2z_3=0$. This is a contradiction; thus, $z_1+z_3+z_1z_2z_3\neq0$ in $X_{0}(\sigma_1^3\cdot \Delta; w_{0})$. In consequence, $X_{0}(\sigma_1^3\cdot \Delta; w_{0})$ is isomorphic to the open subset

$$X_{0}(\sigma_1^3\cdot \Delta; w_{0})=\{(z_1,z_2,z_3)\in\C^3:(z_1+z_3+z_1z_2z_3)\neq0\}\sse\C^3,$$

since the coordinate $z_4$ can be obtained uniquely from any points $(z_1,z_2,z_3)\in\C^3$ in this subset. This shows that $X_{0}(\sigma_1^3\cdot \Delta; w_{0})$ is smooth.

In fact, this provides a rather simple description for this braid variety: it is the open set foliated by the smooth hypersurfaces $(z_1+z_3+z_1z_2z_3)=a$, $a\in\C^*$, which are $($in$)$famous for a variety of reasons, including their appearance in the isomonodromic problem for Painlevé I with a 5/2-singularity at infinity, the study of the augmentation variety for the trefoil, and mirror symmetry for the 2-torus with two transversal Lagrangian 2-disks attached [15, 18], being an instance of a self-mirror affine surface. In particular, an arboreal Lagrangian 3-skeleton for the Stein 3-manifold $X_{0}(\sigma_1^3\cdot \Delta; w_{0})$ is depicted in Figure <ref>. $\Box$

An arboreal Lagrangian 3-skeleton for the Stein structure of the (complex) 3-dimensional braid variety $X_{0}(\sigma_1^3\cdot \Delta; w_{0})$ associated to the trefoil braid $\beta=\sigma_1^3\in \Br_2^+$. The skeleton consists of a circle $\S^1$ times a 2-torus $\mathbb{T}^2$ with two 2-disks attached to it along the curves $(1,0),(0,1)\in H_1(\mathbb{T}^2,\Z)$.

In the case of positive braids associated to algebraic knots $K\sse\R^3$, the braid varieties can be similarly described symplectically using the arboreal skeleta constructed in [15]. In general, following the lines of Example <ref>, the braid varieties for $(2,n)$-torus links can be understood rather immediately, their computation often reducing to the understanding of a certain hypersurface.$\Box$

Note that the braid variety $X_{0}(\sigma_1^3\cdot \Delta; w_{0})$ in Example <ref> contains an $\S^1$-factor in its Lagrangian skeleton, and the Stein surface $X=\{z_1+z_3+z_1z_2z_3=1\}\sse\C^3$ is described by the Lagrangian 2-skeleton given by Figure <ref> without the $\S^1$-factor. Since we can write

$$X_{0}(\sigma_1^3\cdot \Delta; w_{0})\cong\{(z_1,z_2,z_3,t): (z_1+z_3+z_1z_2z_3)t=1\}\sse\C^3\times\C^*,$$

there exists a $\C^*$-action on $X_{0}(\sigma_1^3\cdot \Delta; w_{0})$ whose quotient yields the affine surface $X$. The feature of admitting certain (complex) torus actions with interesting quotients is a general property of braid varieties, as we will now see.

§.§ Torus actions on braid varieties
Let $[\beta]\in \Br^+_n$ be a positive braid with a fixed positive braid word $\beta$. Consider the (Cartan) subgroup $\mathbb{T}\cong(\C^*)^n\sse\GL(n,\C)$ of diagonal matrices. In this section we construct an algebraic $T$-action on the braid variety $X_{0}(\beta)$. First, we observe that

\begin{equation}
\label{eq: slide diagonal}
\displaystyle\left(\begin{matrix} t_1 & 0\\ 0 & t_2\\ \end{matrix}\right)\left(\begin{matrix} 0 & 1\\ 1 & z\end{matrix}\right)=\left(\begin{matrix}0 & 1\\
1 & \frac{t_2}{t_1}z\\ \end{matrix}\right)\left(\begin{matrix} t_2 & 0\\ 0 & t_1\\ \end{matrix}\right).
\end{equation}

Let $D_{\bt}=\diag(t_1,\ldots,t_n)\in\mathbb{T}$ be a diagonal matrix. In general, we have $D_{\bt}B_i(z)=B_i\left(\frac{t_{i+1}}{t_i}z\right)D_{s_i(\bt)}$, for $s_i$ the Coxeter projection of $\sigma_i$. Thus

\begin{equation}
\label{eq:slide diagonal 2}
D_{\bt}B_{i_1}(z_1)\cdots B_{i_r}(z_r)=B_{i_1}(c_1z_1)\cdots B_{i_r}(c_rz_r)D_{w(\bt)},
\end{equation}

where $r=\ell(\beta)$, $c_k=t_{w_k(i_{k}+1)}t^{-1}_{w_k(i_{k})}$, $w_k= s_{i_1}\cdots s_{i_{k-1}}$ and $w=w_{r+1}$ is the permutation corresponding to $\beta$. The torus actions we study are defined as follows:

Let $\beta$ be a positive $n$-braid word of length $r=\ell(\beta)$. The torus $\mathbb{T}$-action of $\mathbb{T}\cong(\C^{*})^{n}$ on affine space $\C^{\ell(\beta)}$ is given by

\bt.(z_1,\ldots,z_r):=(c_1z_1,\ldots,c_rz_r),\quad \bt\in\mathbb{T}, \quad(z_1,\ldots,z_r)\in\C^r,

where $c_i$ are defined as above, $i\in[1,r]$. Note that this torus $\mathbb{T}$-action preserves the braid variety $X_{0}(\beta)\sse\C^{r}$ thanks to relation <ref>.

By definition, the $T$-torus action $T\times X_{0}(\beta)\lr X_{0}(\beta)$ on the braid variety $X_{0}(\beta)$ is the quotient of the restriction of the above $\mathbb{T}$-action to $X_{0}(\beta)$ by the diagonal subtorus $\C^*\sse\mathbb{T}$, i.e. $T:=\mathbb{T}/\C^*_{diag}\cong (\C^*)^{n}/\C^*\cong(\C^*)^{n-1}$. The $\mathbb{T}$-action descends to the $T$-action quotient since the diagonal subtorus $(t,\ldots,t)\sse\mathbb{T}$ acts trivially on $X_{0}(\beta)$.$\Box$

The torus action on $\C^r$ in Definition <ref> depends on the choice of braid word $\beta$. Nevertheless, if $\beta$ and $\beta'$ are two positive presentations of the same braid, $[\beta]=[\beta']\in \Br^+_n$, then there exists an algebraic isomorphism $X_{0}(\beta)\cong X_{0}(\beta')$ which is equivariant with respect to this torus action. In addition, the $T$-action preserves the product decomposition
$$X_{0}(\beta\cdot\FT) \cong X_{0}(\beta\cdot \Delta; w_0)\times \C^{\binom{n}{2}}$$
established in Theorem <ref>. Let us study this $T$-action on braid varieties $X_0(\beta)$.

Let $c(\beta)$ be the number of cycles in the cycle decomposition of the Coxeter projection $\pi(\beta)\in S_n$, i.e. the number of cycles of $\beta$ understood as a permutation. The braid $[\beta]\in \Br_n$ closes up[Either through the rainbow or $(-1)$-framed closure.] to a knot in $\R^3$ if and only if $c(\beta)=1$, and there are $(n-1)!$ such permutations $\pi(\beta)\in S_n$. For a braid associated to a knot, we have the following result:

Let $\beta$ be a positive braid word, $[\beta]\in \Br_n^+$, with $c(\beta)=1$. Then the torus $T$-action of $T\cong(\C^*)^{n-1}$ on the braid variety $X_{0}(\beta)$ is free.

Let $(z_{1}, \dots, z_{r}) \in X_{0}(\beta)$ and assume $\bt.(z_{1}, \dots, z_{r}) = (z_{1}, \dots, z_{r})$ for some $\bt \in (\C^{*})^{n}$. In particular, we have that $B_{\beta}(z) = B_{\beta}(\bt. z)$. Thanks to Equality (<ref>), we have that $D_{\bt}B_{\beta}(z)D_{w(\bt)}^{-1} = B_{\beta}(z)$. Since $z \in X_{0}(\beta)$, the matrix $B_{\beta}(z)$ is upper triangular, and therefore its diagonal entries must be nonzero, as $\det(B_{\beta}(z)) = \pm 1$. From the equation

$$D_{\bt}B_{\beta}(z)D_{w(\bt)}^{-1} = B_{\beta}(z),$$

it follows that $t_{i}t_{w(i)}^{-1} = 1$ for every $i = 1, \dots, n$. Given that $c(\beta)=1$, we must have that $t_{i} = t_{j}$ for all $i, j$ and the result follows.

Let $\beta$ be a positive braid word, $[\beta]\in \Br_n^+$, with $c(\beta)=1$. Then the quotients of the braid varieties $X_{0}(\beta \cdot \FT)/T$ and $X_{0}(\beta \cdot \Delta; w_{0})/T$ are smooth and of dimension $\ell(\beta) + \binom{n}{2} - n +1$ and $\ell(\beta) - n + 1$, respectively.$\Box$

The hypothesis $c(\beta)=1$ in Lemma <ref> is needed, as the $T$-actions on the braid varieties will in general fail to be free. For instance, consider the 2-stranded braid $\beta=\sigma_1^4$, associated to the Hopf link under the $(-1)$-framed closure, and its braid variety

The torus $T$-action scales $z_1$ and $z_3$ by $t\in T\cong\C^*$, and scales $z_2$ and $z_4$ by $t^{-1}$. Hence, it has a fixed point $(z_1,z_2,z_3,z_4)=(0,0,0,0)\in X_{0}(\beta)$. The following remark explains how to proceed in the case that $c(\beta)\neq1$.

Consider a positive braid word $\beta$ such that $[\beta]\in \Br_n^+$ closes up to a link with $k$ connected components, i.e. $c(\beta)=k$. Then, we can construct several subtori of $T$ that act freely on $X_{0}(\beta)$ as follows. Let $O_{1}, \dots, O_{k}$ be the orbits for the action of $w$ on $\{1, \dots, n\}$, where $w$ is the permutation associated to $\beta$. Choose a representative $j_{m} \in O_{m}$ for every $m = 1, \dots, k$. Let $T' \subseteq T$ be the torus defined by the extra equations $t_{j_{1}} = t_{j_{2}} = \cdots = t_{j_{k}}$, noting that $T' \cong (\C^{*})^{n-k}$. The same argument as in the proof of Lemma <ref> shows that $T'$ acts freely on $X_{0}(\beta)$. Note that we obtain that the quotient braid variety $X_{0}(\beta\cdot \Delta; w_{0})/T'$ is a smooth variety of dimension $\ell(\beta) - n + k$, and a similar result holds for the quotient $X_{0}(\beta\cdot\FT)/T'$.$\Box$

This concludes the discussion on the torus $T$-action on $X_0(\beta)$. The geometric structures discussed during the article, e.g. stratifications and holomorphic symplectic structures, are compatible with these torus $T$-actions, and will be studied for the braid varieties $X_0(\beta)$ and their quotients $X_0(\beta)/T$.

§.§ Toric charts in braid varieties

In this subsection, we construct a codimension-$0$ toric chart $T_{\tau(\beta)}\sse X_{0}(\beta \cdot \Delta;w_{0})$ associated to an (arbitrary) ordering $\tau(\beta)\in S_{l(\beta)}$ of the crossings of the positive braid word $\beta$. For that, consider two $n$-braid words
$$\beta=\sigma_{i_1}\sigma_{i_2}\cdot\ldots\cdot\sigma_{i_{k-1}}\sigma_{i_k}\sigma_{i_{k+1}}\cdot\ldots\cdot\sigma_{i_l},\quad  \beta'=\sigma_{i_1}\sigma_{i_2}\cdot\ldots\cdot\sigma_{i_{k-1}}\sigma_{i_{k+1}}\cdot\ldots\cdot\sigma_{i_l},$$

i.e. $\beta'$ is obtained from $\beta$ by removing the $k$th crossing $\sigma_{i_k}$. We will construct a rational map $X_{0}(\beta\cdot\FT) \dashrightarrow X_{0}(\beta'\cdot\FT) \times \C^{*}$ that identifies the latter variety with an explicit open set in $X_{0}(\beta\cdot\FT)$.

First, we start with the following simple:

Let $L$ and $U$ be two invertible lower- and upper-triangular matrices, respectively. Then there exist lower- and upper-triangular matrices $\widetilde{L}$ and $\widetilde{U}$ such that

B_i(z)U=\widetilde{U}B_i\left(\frac{u_{i+1,i+1}z+u_{i,i+1}}{u_{i,i}}\right),\ LB_i(z)=B_i\left(\frac{u_{i+1,i+1}z+l_{i+1,i}}{l_{i,i}}\right)\widetilde{L}.

Moreover, $\widetilde{u}_{i,i+1}=\widetilde{l}_{i+1,i}=0$ and $\widetilde{u}_{k,k} = u_{s_{i}(k), s_{i}(k)}$ for every $k$.

We prove the statement for the upper-triangulation matrix $U$, the case of $L$ is proven analogously. By setting $\widetilde{u}_{i,i+1}=0$, one obtains the system of equations:

\widetilde{u}_{j,i}=u_{j,i+1},\ \widetilde{u}_{j,i}+\widetilde{u}_{j,i+1}(z+u_{i,i+1})=u_{j,i+1},\ (j<i),

\widetilde{u}_{i,j}=u_{i+1,j},\ \widetilde{u}_{i+1,j}=u_{i,j}+zu_{i+1,j},\ (j>i+1).

This system can be readily solved. Indeed, we first deduce that $\widetilde{u}_{k,k} = u_{k,k}$ if $k \neq i, i+1$. From the equations $u_{i,i} + zu_{i+1, i} = \widetilde{u}_{i+1,i+1}$, we obtain $\widetilde{u}_{i+1,i+1} = u_{i, i}$. Finally, one concludes that $\widetilde{u}_{i,i} = u_{i+1, i+1}$.

They key algebraic equality that incarnates opening a crossing $\sigma_i$ in a positive braid word, in terms of braid matrices, reads

\begin{equation}
\label{eq: factorization}
\end{equation}

where the variable $z \in \C^{*}$ associated to that crossing $\sigma_i$ is now assumed to be non-zero. In this equation, we have used the matrices

U_i(z):=\left(\begin{matrix} 1 & z^{-1}\\ 0 & 1\end{matrix}\right),\quad D_i(z):=\left(\begin{matrix} -z^{-1} & 0\\ 0 & z\end{matrix}\right),\quad
L_i(z):=\left(\begin{matrix} 1 & 0\\ z^{-1} & 1\end{matrix}\right),

understood as being the $(2\times 2)$-block matrices placed in $i$-th and $(i+1)$-st row and column. Let us now illustrate how the process of opening a crossing occurs at the level of general braid matrices, as follows. Consider the positive braid word $\beta=\beta_1 \sigma_i \beta_2$ and the braid word $\beta'=\beta_1\beta_2$ obtained by opening (i.e. removing) the explicit crossing $\sigma_i$ between $\beta_1,\beta_2$. In order to apply Equation <ref> we must assume that the variable $z$ associated to the crossing $\sigma_i$ is non-vanishing, and we always do so. Then we write

B_{\beta} = B_{\beta_1}(z_{1}, \dots, z_{r-1})B_i(z)B_{\beta_2}(z_{r+1}, \dots, z_{\ell})=B_{\beta_{1}}U_i(z)D_i(z)L_i(z)B_{\beta_{2}},

and use both Equation (<ref>) and Lemma <ref> to slide the middle matrices to the sides, $U,D$ to the left and $L$ to the right. This results in a decomposition of the form

B_{\beta}=U'D'B_{\beta_{1}}(z'_{1}, \dots, z'_{r-1})B_{\beta_{2}}(z'_{r+1}, \dots, z'_{\ell})L'=U'D'B_{\beta'}(z'_{1}, \dots, z'_{r-1}, z'_{r+1}, \dots, z'_{\ell})L'

where $U'$, $L'$ and $D'$ are some explicit upper (lower) uni-triangular and diagonal matrices, respectively, and $z'_{1}, \dots, z'_{r-1}, z'_{r+1}, \dots, z'_{\ell}$ are polynomial functions on $z_{1}, \dots, z_{r-1}, z_{r}^{\pm 1}, z_{r+1}, \dots, z_{\ell}$. Note that $B_{\beta}(z)L_{1}$ is upper-triangular for some lower-triangular matrix $L_{1}$ if and only if $B_{\beta'}(z')L'L_{1}$ is upper-triangular. These are the first ingredients for the construction of the rational map $\Omega_{\sigma_{i}}: X_{0}(\beta\FT) \dashrightarrow X_{0}(\beta'\FT) \times \C^{*}$.

For the second ingredient, we consider a point $(z_{1}, \dots, z_{\ell}, c_{ij}) \in X_{0}(\beta\cdot\FT)$. By Theorem <ref>, this is equivalent to $B_{\beta}(z)L(c_{ij})$ being upper triangular. Now we open a crossing, so we assume $z_i\neq0$ is non-vanishing: using the decomposition above we obtained that $B_{\beta'}(z'_{1}, \dots, z'_{r-1})L'L(c_{ij})$ is upper triangular. Since $L'L(c_{ij})$ is lower triangular with $1$'s in the diagonal, we can write $L'L(c_{ij}) = L(c'_{ij})$, where $c'_{ij}$ are polynomial functions on $z_{r}^{-1}, z_{r+1}, \dots, z_{\ell}, c_{ij}$. These polynomial functions are the second ingredient. In summary, we obtain:

Consider the positive braid word $\beta=\beta_1 \sigma_i \beta_2$ of length $\ell=l(\beta)$, $\beta'=\beta_1\beta_2$, and suppose that the complex variable $z_i$ associated to the (middle) crossing $\sigma_i$ is non-vanishing. By definition, the rational map $\Omega_{\sigma_{i}}$ associated to opening the crossing $\sigma_i$ is

\Omega_{\sigma_{i}}: X_{0}(\beta\FT) \dashrightarrow X_{0}(\beta'\FT) \times \C^{*}, (z_{1}, \dots, z_{\ell}, c_{ij}) \mapsto (z'_{1}, \dots, z'_{r-1}, z'_{r+1}, \dots, z'_{\ell}, c'_{ij}, z_{r}^{-1}),

where $z_i'\in\C[z_1,\ldots,z_{r-1},z_r^{\pm},z_{r+1},\ldots,z_{\ell}],c_{ij}'\in\C[z_r^{-1},z_{r+1},\ldots,z_\ell,c_{ij}]$ are the polynomial functions defined as above.$\Box$

In the same notation and hypothesis as above, we have that:

The rational map

$$\Omega_{\sigma_{i}}: X_{0}(\beta\FT) \dashrightarrow X_{0}(\beta'\FT) \times \C^{*}$$

restricts to an isomorphism between the open locus $\{z_{r} \neq 0\}\sse X_{0}(\beta\cdot\FT)$ and $X_{0}(\beta'\cdot\FT) \times \C^{*}$.

From the construction, see e.g. Lemma <ref>, if we know $z'_{1}, \dots, z'_{r-1}$, $z'_{r+1}, \dots, z'_{\ell}$ and $z_{r}$ then we can reconstruct $z_{1}, \dots, z_{\ell}$, provided $z_{r} \neq 0$. It remains to show that, if we also know $c'_{ij}$ then we can reconstruct $c_{ij}$ as well. For that, we just notice that we can reconstruct $L'$, and we have the equation $L(c_{ij}) = (L')^{-1}L(c'_{ij})$.

There are two fundamental properties of these rational maps $\Omega_{\sigma_{i}}$: they can be iterated, and they are compatible with the torus $T$-action. This is summarized in the following

Let $\beta$ be a positive $n$-braid word. For each ordering  $\tau(\beta)\in S_{\ell(\beta)}$ of the crossings of $\beta$, there exists an open set $T_{\tau(\beta)}\sse X_{0}(\beta\cdot\FT)$ such that:

(i) $T_{\tau(\beta)}\cong(\C^{*})^{\ell(\beta)} \times X_{0}(\FT) = (\C^{*})^{\ell(\beta)} \times \C^{\binom{n}{2}}$.

(ii) $T_{\tau(\beta)}$ is given by the nonvanishing of Laurent polynomials in $z_{r_{1}}, z'_{r_{2}}, z''_{r_{3}}, \dots, z^{(\ell - 1)}_{r_{\ell}}$; these latter variables can be taken as coordinates of the $(\C^{*})^{\ell(\beta)}$-factor.

(iii) $T_{\tau(\beta)}$ is stable under the torus $(\C^{*})^{n-1}$-action on $X_{0}(\beta \cdot \FT)$.

Given the discussion prior to the statement, the only assertion that needs a proof is the stability under the torus action. For that, we need to show that $z_{r_{1}}, z'_{r_{2}}, \dots, z^{(\ell -1)}_{r_{\ell}}$ are all homogeneous under the $(\C^{\times})^{n-1}$-action. This will be proven in Lemma <ref> and Lemma <ref> below[Both lemmas are independent of the intervening material.], and their corresponding analogues in the case of lower-triangular matrices.

Proposition <ref> and the relation between the braid varieties $X_{0}(\beta \cdot \FT)$ and $X_{0}(\beta \cdot \Delta; w_{0})$, as established in Theorem <ref>, readily imply the following result:

Let $\beta$ be a positive $n$-braid word. For each ordering  $\tau(\beta)\in S_{\ell(\beta)}$ of the crossings of $\beta$, there exists an open set $T_{\tau(\beta)}\sse X_{0}(\beta\cdot \Delta; w_{0})$ which is isomorphic to a torus $T_{\tau(\beta)}\cong(\C^{*})^{\ell(\beta)}$ and stable under the torus $(\C^{*})^{n-1}$-action $X_{0}(\beta \cdot \Delta; w_{0})$.$\Box$

The union of the toric charts $T_{\tau(\beta)}$ in Corollary <ref>, as $\tau(\beta)\in S_{\ell(\beta)}$ ranges through all the possible orderings, does not necessarily cover the entire variety $X_{0}(\beta \cdot \Delta; w_{0})$. Fortunately, we can show that it does cover up to codimension-2:

Let $\beta$ be a positive braid word. The complement

$$X_{0}(\beta \cdot \Delta;w_{0})\setminus\left(\bigcup_{\tau(\beta)\in S_{\ell(\beta)}}T_{\tau(\beta)}\right)\sse X_{0}(\beta \cdot \Delta;w_{0})$$

has codimension at least $2$.

Let us prove this by induction on the length $\ell(\beta)\in\N$. The base case, $\ell(\beta) = 0$ holds, as $X_{0}(\Delta; w_{0}) = \{\operatorname{pt}\}$, see Example <ref>. Note that for the case $\ell(\beta) = 1$, $\beta = \sigma_{i}$ for some $i\in[1,n-1]$, and $X_{0}(\beta \Delta; w_{0})$ is defined by the condition that $B_{i}(z)^{-1}$ admits an $LU$-decomposition. (See the proof of Theorem <ref>.) This is equivalent to the non-vanishing $z \neq 0$ and thus $X_{0}(\beta\cdot\Delta; w_{0}) = \C^*$; thus the statement also holds.

For the induction step, we assume the statement to be true for length $l\in\N$ and suppose that $\ell(\beta) = \ell + 1$. Let $U_{1} := \{z_{1} \neq 0\}$ and $U_{2} := \{z_{2} \neq 0\}$ and let $\beta', \beta''$ be the braids we obtain by opening the first and second crossings of $\beta$, respectively. In particular, $U_{1} = X_{0}(\beta' \cdot \Delta; w_{0}) \times \C^{*}$ and $U_{2} = X_{0}(\beta'' \cdot \Delta; w_{0}) \times \C^{*}$. By the induction assumption, $U_{1}$ and $U_{2}$ can be covered up to codimension-2 by opening crossings in the positive braids $\beta', \beta''$, respectively. Moreover, the complement of $U_{1} \cup U_{2}$ is $\{z_{1} = 0\} \cap \{z_{2} = 0\}$, which has codimension $2$, and the required result follows.

The toric charts $T_{\tau(\beta)}\sse X_{0}(\beta\cdot \Delta; w_{0})$ used in Corollary <ref> and Theorem <ref> are constructed in Proposition <ref>, whose proof we now complete.

§.§ Proof of Proposition <ref>
Let us state and prove Lemma <ref> and Lemma <ref>, which will conclude Proposition <ref>. For that, we use of the following notion:

Let $T$ be the torus $(\C^{*})^{n-1} = (\C^{*})^{n}/\C^*$, and identify its character lattice with $\mathfrak{X}(T) = \{(a_{1}, \dots, a_{n}) \in \Z^{n} \mid \sum a_{i} = 0\}$. Assume that $T$ acts rationally on a domain $R$, and let $U = (u_{a,k}) \in M_{n}(R)$, $w \in S_{n}$. By definition, a matrix $U$ is said to be $w$-admissible if $u_{a,k}$ is homogeneous of weight

\we{u_{a,k}} = \be_{w(a)} - \be_{w(k)}

for every $a, k \leq n$.$\Box$

(Characterization of $w$-admissibility)
Note that we have three natural actions of the torus $T$ on $M_{n}(R)$:

* Component-wise: $(\bt.U)_{a,k} = \bt.u_{a,k}$. (This is an algebra action.)

* By left multiplication: $(\bt U)_{a,k} = t_{a}.u_{a,k}$.

* By right multiplication: $(U\bt)_{a,k} = t_{k}.u_{a,k}$.

Then, $U$ is $w$-admissible if and only if $\bt.U = \bt^{w}U(\bt^{w})^{-1}$.$\Box$

Before we proceed with Lemma <ref>, here is an example of a $w$-admissible matrix:

Let $w \in S_{n}$ be any permutation and assume $z_{0}$ is an invertible element of weight $\we{z_{0}} = \be_{w(i+1)} - \be_{w(i)}$. Then the matrix $U_{i}(z_{0}) = Id + z_{0}^{-1}E_{i, i+1}$ is $w$-admissible.$\Box$

Consider a $w$-admissible matrix $U\in M_n(R)$ and $z$ an element of weight $\we{z} = \be_{w(a)} - \be_{w(b)}+ \be_{w(m)} - \be_{w(k)}$ for some $a, b, m, k = 1, \dots, n$. Then the element $u_{a,k} + zu_{b, m}$ is homogeneous. The salient property of admissible matrices, which motivates their definition, is that they allow us to construct homogeneous elements for the torus action, as the following result shows.

Let $w \in S_{n}$ be a permutation, $U^{0}$ be an invertible upper-triangular $w$-admissible matrix, and $\beta = \sigma_{i_{\ell}}\cdots \sigma_{i_{1}}$ a positive braid word. Consider algebraically independent variables $z_{\ell}, \dots, z_{1}$ with weights

\we{z_{k}} = -\be_{w_{k-1}(i_{k} + 1)} + \be_{w_{k-1}(i_{k})}.

where $w_{d} = ws_{i_{1}}\cdots s_{i_{d}}$, and inductively define[See Lemma <ref>.] the upper triangular matrices $U^{1}, \dots U^{\ell}$ and elements $z_{\ell}', \dots, z_{1}' \in R[z_{\ell}, \dots, z_{1}]$ by the equation

B_{i_{d+1}}(z_{d+1})U^{d} = U^{d+1}B_{i_{d+1}}(z'_{d+1}),

Then the following two facts holds:

(a) The elements $z'_{1}, \dots, z'_{\ell+1}$ are all homogeneous with respect to the torus $T$-action and, moreover, $\we{z'_{d}} = \we{z_{d}}$ for every $d = 1, \dots, \ell$.

(b) For every $d = 0, \dots, \ell$, the matrix $U^{d}$ is invertible, upper triangular, $w_{d}$-admissible and has entries in the polynomial ring $R[z_{d-1}, \dots, z_{1}]$.

The matrices $U^{0}, \dots, U^{\ell}$ are readily invertible upper triangular. In order to prove the remaining claims, we induct on the length $\ell$, with the base $\ell = 0$ holding by assumption.

For the inductive step, suppose that the statement holds for positive braids of length $\ell$, and consider a positive braid $\beta = \sigma_{i_{\ell + 1}}\sigma_{i_{\ell}}\cdots \sigma_{i_{1}}$ of length $\ell + 1$. Note that the matrices $U^{0}, U^{1}, \dots, U^{\ell}$ and the elements $z'_{1}, \dots, z'_{\ell}$ coincide with those for the braid $\sigma_{i_{\ell}}\cdots \sigma_{i_{1}}$, so we only need to show that the element $z'_{\ell + 1}$ is homogeneous of the same weight as $z_{\ell + 1}$, and that the matrix $U^{\ell + 1}$ is $ws_{i_{1}}\cdots s_{i_{\ell+1}}$-admissible. To ease the notation, we will write $i := i_{\ell + 1}$.

By the comment preceding the lemma, each of the entries of the matrix $B_{i}(z_{\ell + 1})U^{\ell}$ is homogeneous. The $(i+1, i+1)$-entry of this matrix is  $u^{\ell}_{i+1, i+1}z_{\ell + 1} + u^{\ell}_{i, i+1}$. Dividing by $u^{\ell}_{i,i}$  we obtain that

z'_{\ell + 1} = \frac{u^{\ell}_{i+1, i+1}z_{\ell+1} + u^{\ell}_{i, i+1}}{u^{\ell}_{i,i}}

is homogeneous. Since the diagonal entries of $U^{\ell}$ have weight $0$ and every entry of $U^{\ell}$ is algebraically independent with $z_{\ell+1}$, we obtain that $z'_{\ell+1}$ is homogeneous of the same weight as $z_{\ell+1}$. Moreover, using again the $w_{\ell}$-admissibility assumption for $U^{\ell}$ we have that every entry of the matrix $U^{\ell}B^{-1}(z'_{\ell+1})$ is homogeneous.

Now $U^{\ell + 1} = B_{i}(z_{\ell + 1})U^{\ell}B^{-1}_{i}(z'_{\ell + 1})$. We check that this matrix is $w_{\ell+1} = w_{\ell}s_{i}$-admissible. Indeed, computing $(\bt^{w_{\ell+1}})U^{\ell+1}(\bt^{w_{\ell+1}})^{-1}$ we have

\begin{array}{rl}
\bt^{ws_{i}}B_{i}(z_{\ell + 1})U^{\ell}B^{-1}_{i}(z'_{\ell+1})(\bt^{ws_{i}})^{-1} = & (\bt. B_{i}(z_{\ell+1}))\bt^{w}U^{\ell}(\bt^{w})^{-1} (\bt.B_{i}^{-1}(z'_{\ell+1})) \\
= & (\bt. B_{i}(z_{\ell+1}))(\bt. U^{\ell})(\bt.B_{i}^{-1}(z'_{\ell+1})) \\
= & \bt.(U^{\ell+1})
\end{array}

where the first equality follows from (<ref>). This concludes the proof thanks to Remark <ref>.

The proof of the following result is similar to that of Lemma <ref> and left to the reader:

Let $U\in M_n(R)$ be a $w$-admissible  upper-triangular matrix and $z_0\in R$ homogeneous and invertible with weight $\we{z_0} = -\be_{w(i+1)} + \be_{w(i)}$. Then the matrix $U' = D_{i}(z_0)UD_{i}^{-1}(z_0)$ is $ws_{i}$-admissible.$\Box$

This concludes the necessary ingredients for Proposition <ref>, and thus completes our argument for Corollary <ref> and Theorem <ref>. The following three subsections relate the results and constructions of Subsections <ref>, <ref> and <ref> to character varieties, through the work of A. Mellit [68], augmentation varieties, as featured in [53, 54], and open Bott-Samelson cells, according to [85, 87].

§.§ Mellit's chart

In this subsection we recast a construction by A. Mellit in the light of braid varieties, in particular defining a certain toric chart in $X_0(\beta\Delta,w_0)$, which we refer to as the Mellit chart. The main result of the subsection is that the Mellit chart can be obtained by our opening-crossing procedure from Subsection <ref> above. In order to connect to [68], we need the following preliminary discussion.

Let $w\in S_n$ be a permutation and $C_{w}=\SB w\SB\sse \GL(n,\C)$ the Bruhat cell corresponding to $w$, where $\SB\sse\GL(n,\C)$ is the Borel subgroup of upper-triangular matrices. Recall that the product of any two matrices in $C_u$ and $C_v$ belongs to $C_{uv}$ if $\ell(uv)=\ell(u)+\ell(v)$. Consequently, for any reduced expression $u=s_{i_1}\cdots s_{i_\ell}$, the associated braid matrix $B_u(z_1,\ldots,z_{\ell})$ belongs to the Bruhat cell $C_{u}$.[Recall that we interchangeably use the notation $s_i$ and $\sigma_i$ for the Artin generators of the braid group, which is particularly well-suited when comparing to the notation used in [68].]

Let $u=s_{i_1}\cdots s_{i_\ell}$ be a reduced expression and suppose that $\ell(us_i)=\ell(u)-1$. Then there exists $k\in\N$ such that:

(a) The matrix $B_u(z_1,\ldots,z_{\ell})B_i(z)$ belongs to the Bruhat cell $C_{u}$ if and only if $z_{k}\neq 0$,

(b) In case $z_k\neq 0$, we can uniquely write $B_u(z_1,\ldots,z_{\ell})B_i(z)=UB_{u}(z'_1,\ldots,z'_{\ell})$
for a certain upper-triangular matrix $U$.

Since $\ell(us_i)=\ell(u)-1$, there exists $k\in\N$ such that $us_i=s_{i_1}\cdots\widehat{s_{i_k}}\cdots s_{i_\ell}$.[This is known as exchange property for the Coxeter group $S_n$.] That is, we can write $u=u_1s_{i_k}u_2$ such that $s_{i_k}u_2=u_2s_{i}$, and thus $us_i=u_1s_{i_k}u_2s_i=u_1s_{i_{k}}s_{i_{k}}u_2=u_1u_2$. This implies the following equation for the braid matrices:

where $z', z'_{k+1},\ldots,z'_{\ell}$ are some functions of $z,z_{k+1},\ldots,z_{\ell}$. If $z_{k}\neq 0$, then we can further write $B_{i_k}(z_k)B_{i_k}(z')=UB_{i_k}(z'')$, so

\widetilde{U}B_u(z'_1,\ldots,z'_{k-1},z'',z'_{k+1},\ldots,z'_{\ell})

and the result is in the Bruhat cell $C_{u}$. If instead $z_{k}=0$, then $B_{i_k}(z_k)B_{i_k}(z')$ is upper-triangular, and $B_u(z_1,\ldots,z_{\ell})B_{i}(z)$ is in the Bruhat cell
$C_{u_1u_2}$, which is disjoint from $C_{u}$.

Consider $\beta_1=s_1s_2s_1s_1$ and $\beta_2=s_1s_2s_1s_2$. Then the braid matrix $B_{\beta_1}(z_1,z_2,z_3,z_4)$ is in the Bruhat cell $C_{s_1s_2s_1}$ if and only if $z_3\neq 0$. In contrast, the braid matrix $B_{\beta_2}(z_1,z_2,z_3,z_4)$
is in the Bruhat cell $C_{s_1s_2s_1}$ if and only if $z_1\neq 0$. In both cases the reduced expression is $u=s_1s_2s_1$.$\Box$

The index $k\in\N$ from Proposition <ref> is unique and can be described geometrically, as follows. Draw a braid diagram for $u$, labeling the strands $1$ to $n$ on the right. Since $\ell(us_i)=\ell(u)-1$, the $i$-th and $(i+1)$-st strands intersect somewhere in the diagram for $u$. Given that $u$ is reduced, they intersect exactly once. The index $k$ corresponds to this intersection point.$\Box$

Let us now compare our construction to A. Mellit's [68], with $\beta\Delta=s_{i_1}\cdots s_{i_{\ell+\binom{n}{2}}}$ a positive braid. In <cit.>, A. Mellit defines a sequence of permutations $p_0=1,p_1,\ldots,p_{\ell+\binom{n}{2}}$ by the rules:

(a) If $\ell(p_{k-1}s_{i_k})=\ell(p_{k-1})+1$ then $p_{k}=p_{k-1}s_{i_k}$,

(b) If $\ell(p_{k-1}s_{i_k})=\ell(p_{k-1})-1$ then $p_{k}=p_{k-1}$.

In his terminology, this sequence is a walk which never goes down. Let us now describe the toric chart used in [68]:

Let $\beta$ be a positive $n$-braid word, the Mellit chart $\mathfrak{M}\sse X_0(\beta\Delta,w_0)$ is defined as the locus of $z_1,\ldots,z_s$ such that

\begin{equation}
\label{eq: mellit chart}
B_{i_1}(z_1)\cdots B_{i_{s}}(z_s)\in C_{p_s}\ \mathrm{for\ all}\ s\le \ell+\binom{n}{2}.
\end{equation}

Note that $\mathfrak{M}\sse X_0(\beta\Delta,w_0)$ is codimension-0 and Zariski open in $X_0(\beta\Delta,w_0)$ .$\Box$

At this stage, our Corollary <ref> provides many toric charts $T_{\tau(\beta)}$ for $X_0(\beta\Delta,w_0)$, (surjectively) indexed by orderings $\tau(\beta)\in S_{\ell(\beta)}$ of the crossings. The toric chart $\mathfrak{M}$ introduced in Definition <ref> is also a subset of $X_0(\beta\Delta,w_0)$, and it is thus natural to ask whether $\mathfrak{M}$ is of the form $T_{\tau(\beta)}$ and, if so, for which ordering $\tau(\beta)$ this is the case. This is answered in our next result (and its proof):

Let $\beta$ be a positive braid word. Then there exists an ordering $\tau(\beta)\in S_{\ell(\beta)}$ of the crossings such that $T_{\tau(\beta)}\sse X_0(\beta\Delta,w_0)$ coincides with the Mellit chart $\mathfrak{M}\sse X_0(\beta\Delta,w_0)$.

The ordering $\tau(\beta)$ in which we open the crossings is as follows. First, we find the smallest  $j$ such that $p_{j-1}=p_j$. This means that
$p_{j-1}=s_{i_1}\cdots s_{i_{j-1}}$ is a reduced word and  $\ell(p_{j-1}s_{i_j})=\ell(p_{j-1})-1$. The condition (<ref>)
holds automatically for $s<j$, and  for $s=j$ we can apply Proposition <ref>: there exists some $k<j$ such that
$B_{i_1}(z_1)\cdots B_{i_{j}}(z_j)\in C_{p_j}$ if and only if $z_k\neq 0$.

It follows from Remark <ref> that the crossing with index $k$ is in the braid $\beta$, and never in $\Delta$.
We can open this crossing and obtain a new braid $\beta' \Delta$.  By Proposition <ref>, a point in $X_0(\beta\Delta,w_0)$ is in the Mellit chart if and only if the corresponding point in $X_0(\beta'\Delta,w_0)$ is in the respective chart. This process can be continued iteratively. Eventually, all crossings in $\beta$ will be exhausted, and we reach a reduced expression $\Delta$, which satisfies the defining inclusion (<ref>) automatically.

Consider the positive 3-braid $\beta=\sigma_1\sigma_2\sigma_1$, and thus $\beta \Delta=\sigma_1\sigma_2\sigma_1\sigma_1\sigma_2\sigma_1$. By opening the third crossing $\sigma_1$, from the left, we reach the braid word $\sigma_1\sigma_2\sigma_1\sigma_2\sigma_1$. Then we open the first (leftmost) $\sigma_1$ crossing and obtain $2121$. Finally, opening again the first (leftmost) crossing $\sigma_1$ in the resulting braid (which corresponds to the second crossing in the original braid) we reach the positive braid word $\Delta=\sigma_1\sigma_2\sigma_1$. This sequence of crossings $\tau(\beta)$ yields a toric chart $T_{\tau(\beta)}\sse X_0(\beta\Delta,w_0)$ which coincides with the Mellit chart $\mathfrak{M}\sse X_0(\beta\Delta,w_0)$.$\Box$

This concludes our discussion on the Mellit chart and the relation between our Corollary <ref> and <cit.>. Let us shift our focus towards an class of algebraic varieties which are central to the study of Legendrian links in contact 3-manifolds, augmentation varieties.

§.§ Augmentation varieties as quotient braid varieties
In this subsection, we establish a connection between braid varieties and augmentation varieties. The latter are a class of varieties that feature saliently in the study of Floer-theoretic invariants associated to Legendrian links $\La\sse(\R^3,\xi_\st)$. The reader is referred to [38] for the basics of 3-dimensional contact topology, [70] for an excellent survey on Floer-theoretic invariants of Legendrian knots, and [23, 53, 54] for further details.

Let $\beta \in \mathrm{Br}^{+}_{n}$ be a positive braid word, and $\Lambda(\beta)\sse(\R^3,\xi_\st)$ the Legendrian links associated to the $(-1)$-framed closure of the braid $\beta$, see [16, 20]. Let us also choose a collection of marked (base) points $\ft \subseteq \beta$ on the Legendrian links, following [71, 72]. In our case, the two choices for marked points that we use are: a choice of one marked point per strand of the braid $\beta$, this collection of marked points will be denoted by $\ft_{s}$; a choice of one marked point per component of the Legendrian link $\Lambda(\beta)\sse(\R^3,\xi_\st)$, this collection will be denoted by $\ft_{c}$. By convention, we place any marked points on the strands and to the right of all crossings in $\beta$.[Though not essential, this convention will be useful in simplifying some statements. Note also that $\ft_{c}$ technically depends on a choice of strand per component of $\Lambda(\beta)$, but for the sake of readability we prefer to not include this into our notation.]

Let $\SA(\beta, \ft)$ be the Legendrian Contact DGA of the Legendrian link $\La(\beta)\sse(\R^3,\xi_\st)$, also known as the Chekanov-Eliashberg DGA. This differential graded algebra (DGA) is an invariant of the Legendrian link $\La(\beta)\sse(\R^3,\xi_\st)$ (with marked points) up to Legendrian isotopy. It was defined by Y. Chekanov [23] over $\Z_2$-coefficients, and latter lifted to $\Z$-coefficients and marked points [71, 72], see [70] for a survey. The differential $\SA(\beta, \ft)$ is given by a Floer-theoretical count of (pseudo)holomorphic strips whose asymptotics are governed by the Legendrian link $\La\sse(\R^3,\xi_\st)$. By definition, the augmentation variety $\Aug(\beta, \ft)$ associated to $(\beta, \ft)$ is the affine algebraic variety $\Aug(\beta, \ft):=\mbox{Hom}_{\mbox{dg}}(\SA(\beta, \ft),\C)$ of all DGA morphisms from $\SA(\beta, \ft)$ to the DGA $\C$, the latter being concentrated in degree 0 and with trivial differential.

In the case of Legendrian links $\La\sse(\R^3,\xi_\st)$ associated to positive braids, $\La\simeq\La(\beta)$, augmentation varieties $\Aug(\beta, \ft)$ are closely related braid varieties. This will follow rather simply from the work of T. Kálmán [54], and it is the content of the following:

Let $\beta$ be a positive braid word, $[\beta] \in \mathrm{Br}^{+}_{n}$. The following two statements hold:

(i) There exists an algebraic isomorphism $\Aug(\beta, \ft_{s}) \cong X_{0}(\beta\cdot\Delta; w_{0})$.

(ii) Let $T \subseteq (\C^{*})^{n}$ be the algebraic torus determined by $t_{w^{-1}(i)} = 1$ if the $i$th strand of the braid $\beta$ has a marked point in $\mathfrak{t}_c$. Then, there exists an algebraic isomorphism

\Aug(\beta, \ft_{c}) \cong X_{0}(\beta\cdot\Delta;w_{0})/T.

Let us use the following characterization by T. Kálmán [53, 54] (see also [20]): if $\beta$ is a positive braid word and $i_{1}, \dots, i_{s}$ are strands that carry a marked point (to the right of every crossing) then the augmentation variety is the affine subvariety of $\C^{\ell(\beta) + \binom{n}{2}}\times (\C^{*})^{s}$ given by the equation

\begin{equation}\label{eq:kalman}
B_{\beta}(z)\left(\begin{matrix} 1 & 0 & \cdots & 0 \\ c_{21} & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ c_{n1} & c_{n2} & \cdots & 1\end{matrix}\right)\diag(t_{1}, \dots, t_{n}) \; \; \text{is upper triangular with a prescribed diagonal}
\end{equation}

where the notation follows the convention that in $\diag(t_{1}, \dots, t_{n})$ we have $t_{i} = 1$ if $i \not= i_{1}, \dots, i_{s}$. For the choice of marked points $\mathfrak{t}_s$, this reduces to $B_{\beta}(z)B_{\Delta}(u)w_{0}$ being upper triangular, which is precisely the definition of $X_{0}(\beta\cdot\Delta; w_{0})$.  This establishes the statement in $(i)$. For the choice of marked points $\mathfrak{t}_c$, as in $(ii)$, Equation (<ref>) reduces to $B_{\beta}(z)B_{\Delta}(u)w_{0}$ being upper triangular with a prescribed diagonal outside of the strands carrying marked points. Since the action of $T$ on $X_{0}(\beta\cdot \Delta; w_{0})$ is free (see Remark <ref>) the quotient map $X_{0}(\beta \cdot \Delta; w_{0}) \to X_{0}(\beta\cdot \Delta; w_{0})/T$ is a principal $T$-bundle. In consequence, $X_{0}(\beta\cdot \Delta; w_{0})/T$ is equivalent to the closed subvariety of $X_{0}(\beta\cdot \Delta; w_{0})$ given by prescribing the diagonal elements in $B_{\beta\cdot\Delta}w_{0}$ at entries corresponding to strands not carrying marked points.

In contact geometry, opening a crossing from $\beta=\beta_1\sigma_i\beta_2$ to $\beta'=\beta_1\beta_2$ can be realized[This is correct in the case that the positive braid has a half-twist remaining [20, 27], which will always be the case in our context.] by an embedded exact Lagrangian cobordism $L_i\sse(\R^3\times\R_t,d(e^t\alpha))$ in the symplectization of $(\R^3,\xi_\st)$, with $\dd L_i=\dd_-L_i\cup\dd_+L_i$ and $\dd_-L_i=\La(\beta')$ and $\dd_+L_i=\La(\beta)$ [1, 10]. It follows from the Floer-theoretic functoriality proven in [27, 75] that such a Lagrangian cobordism induces a map $\Phi_{L_i}:\Aug(\beta',\mathfrak{t})\lr\Aug(\beta,\mathfrak{t})$ between augmentation varieties. It follows from [20, 27] that the ($\Z$-lifted) Floer-theoretical map $\Phi_{L_i}$ agrees with the (quotient of the) map $\Omega_{\sigma_{i}}$ we constructed in Subsection <ref>.

The toric charts we constructed in Corollary <ref>, using Proposition <ref>, can now be used to stratify the augmentation varieties in Theorem <ref>, as follows:

Let $\beta$ be a positive braid word, $[\beta]\in \Br_n^+$, with $c(\beta)=k$.[Geometrically, the Legendrian link has $\La(\beta)$ has $k$ connected components.] For each ordering $\tau(\beta)\in S_{\ell(\beta)}$ of the crossings of $\beta$ there exists codimension-0 toric charts $T^c_{\tau(\beta)}\sse \Aug(\beta, \ft_{c})$ and $T^s_{\tau(\beta)}\sse \Aug(\beta, \ft_{s})$, with $T^c_{\tau(\beta)}\cong(\C^{*})^{\ell(\beta) - n + k}$ and $T^s_{\tau(\beta)}\cong(\C^{*})^{\ell(\beta)}$. In both cases, the complements

$$\Aug(\beta, \ft_{c})\setminus\left(\bigcup_{\tau(\beta)\in S_{\ell(\beta)}}T^c_{\tau(\beta)}\right)\sse \Aug(\beta, \ft_{c}),\qquad \Aug(\beta, \ft_{s})\setminus\left(\bigcup_{\tau(\beta)\in S_{\ell(\beta)}}T^s_{\tau(\beta)}\right)\sse \Aug(\beta, \ft_{s})$$

have codimension at least 2.

In view of Theorem <ref>, only the statements for $\Aug(\beta, \ft_{c})$ remain unproven, but these follow from the $T$-stability of the toric charts on the braid variety $X_{0}(\beta\cdot \Delta; w_{0})$, cf. Corollary <ref>.

§.§ Open Bott-Samelson varieties

This section is not required for the rest of the manuscript: it is provided here for contextual completeness with respect to the articles [16, 37, 85, 86]. The purpose of this section is to relate the braid variety $X_{0}(\beta)$ to the (diagonal) open Bott-Samelson variety $\OBS(\beta)$ associated to the braid $\beta$. This is achieved in Theorem <ref> below, after a brief reminder on Bott-Samelson varieties.

Consider $G := \operatorname{GL}(n,\C)$, $\cB \subseteq G$ the Borel subgroup of upper-triangular matrices and the flag variety $\Fl := G/\cB$. The projective variety $\Fl$ is the moduli space of complete flags of subspaces in $\C^n$: an element $\cF \in \Fl$ is a flag $\cF = (F_1 \subseteq \cdots \subseteq F_n)$ where $\dim F_{i} = i$.
Given a flag $\cF\in \Fl$, we can choose a basis $(v_{1}, \dots, v_{n})$ of $\C^n$ such that $F_{j} = \langle v_{1}, \dots, v_{j}\rangle $ for $ j = 1, \dots, n$; we denote by $V_{\cF} \in G$ the matrix whose columns are the vectors $v_{i}$ expressed in the standard basis. Conversely, given a matrix $V \in G$, we can consider a flag $\cF^{V} = (F_{1} \subseteq \cdots \subseteq F_{n})$ where $F_{j}$ is the span of the first $j$ columns of the matrix $V$. In this correspondence, two flags are equal $\cF^{V} = \cF^{V'}$ if and only if their matrices $V,V'$ are related by an upper triangular matrix, i.e. $V = V'U$ for some $U \in \cB$.

By definition, two flags $\cF,\cF'\in\Fl$ are in relative position $s_{i}$, $i\in[1,n-1]$, if $F_{j} = F'_{j}$ for $j \neq i$ and $F_{i} \neq F'_{i}$. In terms their matrices, the flags $\cF^{V},\cF^{V'}$ are in relative position $s_{i}$ if and only if there exist upper-triangular matrices $A_{1}$ and $A_{2}$ such that $V' = VA_{1}s_{i}A_{2}$, where $s_{i}$ is understood as a permutation matrix.

Since the permutation matrix $s_{i} = B_{i}(0)$ is a braid matrix with the variable set to zero, it follows from Lemma <ref> that the flags $\cF^{V}$ and $\cF^{V'}$ are in relative position $\pi_{i}$ if and only if there exist an upper-triangular matrix $U$ and $z \in \C$ such that $V' = VUB_{i}(z)$.$\Box$

Building on the articles [11, 25], and the subsequent developments [16, 85, 86, 87], we introduce the two algebraic varieties $\OBS(\beta)$ and $\OBS'(\beta)$, as follows:

Let $\beta = \sigma_{i_{1}}\cdots \sigma_{i_{\ell}}$ be a positive braid word.

(i)  The open Bott-Samelson variety $\OBS(\beta) \subseteq \Fl^{\ell+1}$ associated to $\beta$ is the moduli space of $(\ell+1)$-tuples of flags $(\cF_{0}, \dots, \cF_{\ell})$ such that consecutive flags $\cF_{k-1},\cF_{k}$ are in relative position $s_{i_{k}}$, for each $k\in[1,\ell]$.

(ii) The diagonal open Bott-Samelson variety $\OBS'(\beta) \subseteq \OBS(\beta)$ is the closed subvariety defined by the additional condition that $\cF_{0} = \cF_{\ell}$.$\Box$

The diagonal open Bott-Samelson variety $\OBS'(\beta)$ will be related to the braid variety, as we now explain. First, let us construct a map $\pi: G \times X_{0}(\beta) \to \OBS'(\beta)$ as follows. Consider a point $(z_{1}, \dots, z_{\ell}) \in X_{0}(\beta)$ and a matrix $V\in G$, and define $V_{k} := VB_{i_{1}}(z_{1})\cdots B_{i_{k}}(z_{k}) \in G$. The map $\pi$ is then defined by:

\pi: G \times X_{0}(\beta) \lr \OBS'(\beta),\quad \pi(V, z_{1}, \dots, z_{\ell}):= (\cF^{V}, \cF^{V_{1}}, \dots, \cF^{V_{\ell}}).

It follows from Remark <ref> that $\pi(V, z_{1}, \dots, z_{\ell}) \in \OBS(\beta)$ and since $V_{\ell} = VB_{\beta}(z_{1}, \dots, z_{\ell})$, and $(z_{1}, \dots, z_{\ell}) \in X_{0}(\beta)$, we actually have that $\pi(V, z_{1}, \dots, z_{\ell}) \in \OBS'(\beta)$. Thus the image of $\pi$ belongs to $\OBS'(\beta)\sse\OBS(\beta)$, as written above. This map is, in general, not an isomorphism. Nevertheless, we will now construct a right $\cB$-action on the product $G \times X_{0}(\beta)$, and $\pi$ will descend to an isomorphism on the quotient.

Indeed, consider an upper-triangular matrix $U = U^{0} \in \cB$ and define $z_{1}', \dots, z_{\ell}'$ and $U^{1}, \dots, U^{\ell} \in \cB$ inductively via the equation

\begin{equation}\label{eqn: Us}
B_{i_{\ell - k}}(z_{\ell - k})U^{k} = U^{k+1}B_{i_{\ell - k}}(z'_{\ell - k}).
\end{equation}

It follows from the equation $U^{\ell}B_{\beta}(z_{1}, \dots, z_{\ell}) = B_{\beta}(z_{1}', \dots, z_{\ell}')U^{0}$ that $(z_{1}, \dots, z_{\ell}) \in X_{0}(\beta)$ if and only if $(z_{1}', \dots, z_{\ell}') \in X_{0}(\beta)$. For each $(V, z_{1}, \dots, z_{\ell}) \in G \times X_{0}(\beta)$ and upper-triangular matrix $U = U^{0} \in \cB$, we define its (right) action by:

(V, z_{1}, \dots, z_{\ell})\cdot U := (VU^{\ell}, z'_{1}, \dots, z'_{\ell}).

The usefulness of this right action is manifest in the main result of this

Let $\beta$ be a positive braid word, $G=\GL(n,\C)$ and $\cB\sse G$ the Borel subgroup of upper-triangular matrices. Then

(i) The right $\cB$-action on $G \times X_{0}(\beta)$ defined above is free.

(ii) The map $\pi: G \times X_{0}(\beta) \lr \OBS'(\beta)$  induces an isomorphism $(G \times X_{0}(\beta))/\cB \cong \OBS'(\beta)$.

Let us first prove the freeness of the right $\cB$-action. Indeed, suppose that there exists a fixed point, i.e. there exist $U \in \cB$ and $(V, z_{1}, \dots, z_{\ell}) \in G \times X_{0}(\beta)$ such that

$$(V, z_{1}, \dots, z_{\ell})\cdot U = (V, z_{1}, \dots, z_{\ell}).$$

Since $z'_{j} = z_{j}$ for every $j\in[1,\ell]$, it follows from Equation (<ref>) that the matrices $U, U^{1}, \dots, U^{\ell}$ are pair-wise conjugate. In particular, the initial upper-triangular matrix $U$ is conjugate to $U^{\ell}$. Nevertheless, the condition $VU^{\ell} = V$ implies that $U^{\ell} =\mbox{Id}$, and thus no fixed point may exist. The action is thus free.

Second, let us show that the map $\pi$ is surjective, onto the diagonal open Bott-Samelson variety $\OBS'(\beta)$. For that, consider a point $(\cF_{0}, \dots, \cF_{\ell}) \in \OBS'(\beta)$ and let $V \in G$ be any matrix such that $\cF^{V} = \cF_{0}$. Thanks to Remark <ref>, we have that there exist upper-triangular matrices $U^{1}, \dots, U^{\ell}$ and $z_{1}, \dots, z_{\ell} \in \C$  such that
$$\cF_{k} = \cF^{VU_{1}B_{i_{1}}(z_{1})\cdots U_{k}B_{i_{k}}(z_{k})}.$$

Now, use Lemma <ref> to slide all the upper triangular matrices $U_{2}, \dots, U_{\ell}$ to the right; this yields obtain upper-triangular matrices $U'_{1} = U_{1}, U'_{2}, \dots, U'_{\ell}$ and $\hat{z}_{1}, \dots, \hat{z'}_{\ell}$ with the property that

VU'_{1}\cdots U_{\ell}' B_{i_{1}}(\hat{z}_{1})\cdots B_{i_{k}}(\hat{z}_{k}) = VU_{1}B_{i_{1}}(\hat{z}_{1})\cdots U_{k}B_{i_{k}}(\hat{z}_{k})\hat{U},

where $\hat{U}$ is an upper-triangular matrix depending on $k$. This implies that $\pi(VU'_{1}\cdots U'_{\ell}, \hat{z}_{1}, \dots, \hat{z}_{\ell}) = (\cF_{0}, \dots, \cF_{\ell})$. It remains to show that $(\hat{z}_{1}, \dots, \hat{z}_{\ell}) \in X_{0}(\beta)$, that is, the matrix $B_{\beta}(\hat{z}_{1}, \dots, \hat{z}_{\ell})$ is upper-triangular. Since $\cF_{0} = \cF_{\ell}$, the matrices $V$ and $VU'_{1}\cdots U'_{\ell}B_{\ell}(\hat{z}_{1}, \dots, \hat{z}_{\ell})$ differ by an upper-triangular matrix. Since $U'_{1}, \dots, U'_{\ell}$ are upper-triangular, the result follows and $\pi$ is surjective.

Third, let us prove that the map $\pi$ is $\cB$-invariant; we need to check that for every $k$ the matrices $VB_{i_{1}}(z_{1})\cdots B_{i_{k}}(z_{k})$ and $VU^{\ell}B_{i_{1}}(z'_{1})\cdots B_{i_{k}}(z'_{k})$ differ by an upper-triangular matrix. That said, it follows from Equation (<ref>) that this matrix is precisely $U^{\ell - k}$, which proves $\cB$-invariance.

Finally, we must show that if $\pi(V, z_{1}, \dots, z_{\ell}) = \pi(V', z'_{1}, \dots, z'_{\ell})$ then there exist an upper triangular matrix $U$ such that $(V', z'_{1}, \dots, z'_{\ell}) = (V, z_{1}, \dots, z_{\ell})\cdot U$. For that, note that $\cF^{V} = \cF^{V'}$ implies that there exist an upper-triangular matrix, say $U^{\ell}$, such that $V' = VU^{\ell}$. Since $\cF^{VB_{i_{1}}(z_{1})} = \cF^{V'B'_{i_{1}}(z'_{1})}$, there also exist an upper-triangular matrix, say $U^{\ell - 1}$ such that $V'B'_{i_{1}}(z'_{1}) = VB_{i_{1}}(z_{1})U^{\ell - 1}$. In consequence, we obtain the equality $VU^{\ell}B_{i_{1}}(z'_{1}) = VB_{i_{1}}(z_{1})U^{\ell-1}$, and thus $U^{\ell}B_{i_{1}}(z'_{1}) = B_{i_{1}}(z_{1})U^{\ell - 1}$. Note that this is precisely Equation (<ref>). We iterate this procedure until we find $U^{0}$, which is the required upper-triangular matrix. This concludes the proof of the statement.

The concludes our discussion relating braid varieties to open Bott-Samelson varieties. Let us now move to the construction of a holomorphic symplectic structure on braid varieties $X_0(\beta)$.

As a side note, the homotopy types of the varieties $X_0(\beta)$ and $\OBS'(\beta)$ appear to be related to the spectra constructed in [58]. This remains to be explored.$\Box$

§ HOLOMORPHIC SYMPLECTIC STRUCTURE

This section constructs a holomorphic symplectic structure on the (quotient) braid varieties $X_0(\beta\Delta;w_0)/T$, establishing the remainder of Theorem <ref>.(iii). In particular, Theorem <ref> will imply that the augmentation variety associated to a Legendrian link $\La(\beta)$, $\beta$ a positive braid (word), is holomorphic symplectic. In addition, the toric charts we built in Corollary <ref> will actually be (exponential) Darboux charts for this holomorphic symplectic structure. The construction we present draws from the literature on character varieties, where the holomorphic symplectic structures on character varieties have a central role, starting with the Atiyah-Bott-Goldman structures [3, 39] and continuing with, e.g., the work of P. Boalch and L. Jeffrey [6, 7, 52, 68].

§.§ Construction of a 2-form
First, let us review the construction of a 2-form on the braid variety $X_{0}(\beta)$ according to [52, 68], often referred to as the tautological 2-form.
For that, let $\theta:=f^{-1}df$ and $\theta^{R}:=df f^{-1}$ denote respectively the left- and right-invariant algebraic 1-forms on the (complex) Lie group $G=\GL(n,\C)$; these 1-forms are valued in the Lie algebra $\mathfrak{g}=\mathfrak{gl}(n)$, and $\theta$ is referred to as the Maurer-Cartan form. The following two facts can be readily verified:

(a) The 3-form $\Omega:=\Tr(\theta[\theta,\theta])$ is closed and represents a class in $H^3(G)$. This class is in fact dual to the fundamental class of (a copy of) $\S^3\simeq SU(2)\subset GL(n,\C)$.

(b) There is a 2-form $(f|g):=\Tr(\pi_1^*\theta\wedge \pi_2^*\theta^R)=\Tr(f^{-1}df\wedge dg g^{-1})$ on $G\times G$ satisfying the following two “cocycle conditions":

\begin{equation}
\label{eq: cocycle 1}
\end{equation}

\begin{equation}
\label{eq: cocycle 2}
\end{equation}

where $\pi_1,\pi_2,m:G\times G\to G$ are the two projections and the Lie group multiplication map.

Let $X$ be an arbitrary algebraic variety. By definition, a map $f: X\to G$ is $\Omega$-trivial if $f^*\Omega=d\omega$ for some 2-form $\omega$ on $X$. Now, suppose that two maps $f:X\to G$ and $g:Y\to G$ are $\Omega$-trivial, then the product

f\cdot g: X\times Y\xrightarrow{f\times g}G\times G\xrightarrow{m} G

is also $\Omega$-trivial. Indeed, if $f^*\Omega=d\omega_X$ and $g^*\Omega=d\omega_Y$ then (<ref>) implies

(f\cdot g)^*\Omega=d(\omega_X+\omega_Y+(f|g)).

In general, if $f_i:X_i\to G, i\in[1,r]$ are $\Omega$-trivial maps then we can define the form

\omega=\sum \omega_{X_i}+(f_1|f_2)+(f_1f_2|f_3)+\ldots+(f_1\cdots f_{r-1}|f_{r})=

\sum \omega_{X_i}+(f_1|f_2|\cdots|f_r).

The condition (<ref>) implies that this operation defines an associative convolution $(f_1|f_2|\cdots|f_r)$ on collections of $\Omega$-trivial maps. The following identity  will be useful for us, and follows immediately from the definition:

\begin{equation}
\label{eq: form merge}
(f_1|\cdots |f_r)=(f_1|\cdots|f_{j}f_{j+1}|\cdots |f_r)+(f_{j}|f_{j+1}).
\end{equation}

This summarizes the necessary ingredients. Consider a positive braid word $\beta$, with length $r=\ell(\beta)$ and let us apply this construction to our braid varieties $X_0(\beta)$, as follows. The first key fact is that the maps $B_i(z):\C\to G$ given by the braid matrices are $\Omega$-trivial with vanishing primitive 2-form $\omega=0$. Hence, the map $B_{\beta}:\C^{r}\to G$ is also $\Omega$-trivial with primitive being the 2-form

\begin{equation}
\label{eq: def form}
\omega_{\beta}=(B_{i_1}(z_1)|B_{i_2}(z_2))+(B_{i_1}(z_1)B_{i_2}(z_2)|B_{i_3}(z_3))+\ldots+(B_{i_1}(z_1)\cdots B_{i_{k-1}}(z_{k-1})|B_{i_k}(z_{k})).
\end{equation}

The restriction of the 2-form $\omega_{\beta}$ to the braid variety $X_{0}(\beta)$ is closed.

Note that the map $B_{\beta}:X_{0}(\beta)\lr G$  lands in the subgroup of upper-triangular matrices, and the restriction of the 3-form $\Omega$ to the space of upper-triangular matrices vanishes. Therefore, since $d$ commutes with pull-back, we have

i.e. $\omega_{\beta}$ is a closed 2-form.

The following example will prove useful:

Let $\Delta\in \Br_n^+$ be the positive braid $($word$)$ associated to the half-twist; then we have that the 2-form $\omega_{\Delta}=0$ vanishes. $($Thus, it is closed.$)$ Following Lemma <ref>, we can write

B_{\Delta^2}=B_{\Delta}(c)B_{\Delta}(u)=Lw_0\cdot w_0U=LU,

where two copies of $B_{\Delta}$ depend on two sets of independent variables $c_{ij}$ and $u_{ij}$. The 2-form associated to $\Delta^2$ then reads:

\omega_{\Delta^2}=\omega_{\Delta}(c)+(B_{\Delta}(c)|B_{\Delta}(u))+\omega_{\Delta}(u)=(B_{\Delta}(c)|B_{\Delta}(u))=

Consider now the toric charts $T_{\tau(\beta)}\sse X_{0}(\beta\cdot\FT)$ constructed in Corollary <ref>.

Let $\beta$ be a positive braid (word) and $\tau(\beta)\in S_{\ell(\beta)}$. The restriction of 2-form $\omega_{\beta\cdot \Delta^2}$ to any toric chart $T_{\tau(\beta)}\sse X_{0}(\beta\cdot\FT)$ has constant coefficients in the canonical coordinates $($associated to the $D_i$ matrices$)$.

By Lemma <ref>, we can write

B_{\beta\cdot \Delta^2}=B_{\beta}B_{\Delta^2}=B_{i_1}(z_1)\cdots B_{i_r}(z_r)LU.

By Example <ref>, we can also write

\omega_{\beta\cdot \Delta^2}=\omega_{\beta}+(B_{\beta}|B_{\Delta^2})+\omega_{\Delta^2}=

\omega_{\beta}+(B_{\beta}|LU)+(L|U)=
(B_{i_1}(z_1)|\cdots |B_{i_r}(z_r)|L|U).

Next, we need to understand the behavior of the 2-form under opening the crossings according to $\tau(\beta)$, as this determines the construction of the toric chart $T_{\tau(\beta)}$. By using the decomposition in Equation <ref>, we can write

( \cdots|B_{i_s}(z_s)|\cdots )=

( \cdots|U_{i_s}(z_s)|D_i(z_s)|L_i(z_s)|\cdots )-

and clearly $(U_{i_s}(z_s)|D_i(z_s)|L_i(z_s))=0$. Next, assume that $U$ is an upper uni-triangular matrix, then

(\cdots |B_i(z)|U|\cdots)=(\cdots |B_i(z)U|\cdots )+(B_i(z)|U)=

(\cdots |\widetilde{U}B_i(z')|\cdots )+(B_i(z)|U)=
(\cdots |\widetilde{U}|B_i(z')|\cdots )+(B_i(z)|U)-(\widetilde{U}|B_i(z')).

The terms $(B_i(z)|U),(\widetilde{U}|B_i(z'))$ in fact vanish. Indeed, observe that

B_i^{-1}(z)dB_i(z)=\left(\begin{matrix} -z & 1\\ 1 & 0\\ \end{matrix}\right)\left(\begin{matrix} 0 & 0\\ 0 & dz\\ \end{matrix}\right)=\left(\begin{matrix} 0 & dz\\ 0 & 0\\ \end{matrix}\right),

while $dU\cdot U^{-1}$ is upper-triangular, so $(B_i(z)|U)=0$. Similarly,

dB_i(z')\cdot B_i(z')^{-1}=\left(\begin{matrix} 0 & 0\\ 0 & dz'\\ \end{matrix}\right)\left(\begin{matrix} -z' & 1\\ 1 & 0\\ \end{matrix}\right)=\left(\begin{matrix} 0 & 0\\ dz'& 0\\ \end{matrix}\right),

while $\widetilde{U}_{i,i+1}=0$ and hence $(\widetilde{U}|B_i(z'))=0$. We conclude that

(\cdots |B_i(z)|U|\cdots )=(\cdots |\widetilde{U}|B_i(z')|\cdots ),

and similarly $(\cdots |D_i(z)|U|\cdots )=(\cdots |\widetilde{U}|D_i(z)|\cdots ).$
The conclusion from this computation is that the 2-form $\omega_{\beta\cdot \Delta^2}$ does not change as we move $U$ to the left. (Similarly, it does not change as we move lower-triangular matrices to the right.) In conclusion, we are left with several upper uni-triangular matrices, followed by several diagonal matrices and by several lower uni-triangular matrices. Since $(U|U')=(L|L')$ for any pair of upper (resp. lower) uni-triangular matrices, we can consolidate all upper and all lower uni-triangular matrices and write

\omega_{\beta\cdot \Delta^2}=(\widetilde{U}|D_1|\cdots|D_k|\widetilde{L}|U).

Since $\widetilde{U}D_1\cdots D_k\widetilde{L}U$ is upper-triangular, we get $\widetilde{L}=1$. Thus

\omega_{\beta\cdot \Delta^2}=(\widetilde{U}|D_1|\cdots|D_k|U)=(D_1|\cdots |D_k)=\sum_{i<j} \Tr(d\log(D_i)\wedge d\log(D_j)).

and the result follows.

The proof of Lemma <ref> actually shows that the entries of the $D_i$ matrices associated to (opening the crossings for) $\beta$ are quite close to (exponential) Darboux-like coordinates[The term Darboux-like is used here as the form might not be symplectic at this stage.], only differing from an exponential Darboux-like chart by a linear change in constant coefficients. We have now discussed closedness of the 2-form $\omega_{\beta\cdot \Delta^2}$ and its expression in the toric charts $T_{\tau(\beta)}$. In order to show that $\omega_{\beta\cdot \Delta^2}$ induces an holomorphic symplectic structure, as stated in Theorem <ref>.(iii), it suffices to show non-degeneracy, which we now address.

§.§ Non-degeneracy of $\omega_{\beta\cdot \Delta^2}$
Let us consider the Mellit chart $\mathfrak{M}$, as constructed in Theorem <ref>, and $\omega_{\mathfrak{M}}$ its corresponding $2$-form.[Note that the torus $(\mathfrak{M}, \omega_{\mathfrak{M}})$ has already appeared in the work of Mellit <cit.>.] In this subsection, we will first show that $\omega_{\mathfrak{M}/T_{\pi}}$ is non-degenerate, and thus (holomorphic) symplectic. Then we prove, in Theorem <ref>, that $\omega$ induces the holomorphic symplectic structure according to Theorem <ref>.(iii).

Following <cit.>, we can construct a topological avatar for this torus, as follows. Consider a labeled marked surface $(\cS,A,B)$, i.e. an oriented surface $\cS$ with boundary $\partial \cS$ and two sets $A:= \{1, 2, \dots, n\}, B:= \{1', 2', \dots, n'\} \subseteq \partial\cS$ such that:

- Each connected component of $\cS$ has a boundary component.

- Each boundary component intersects both $A$ and $B$.

- The elements of $A$ and $B$ in each boundary component alternate.

Let us denote the two Abelian groups $\Lambda := H_{1}(\cS, A)$ and $\Lambda' := H_{1}(\cS, B)$. Since $A$ and $B$ are alternating, there is a perfect pairing $\cdot: \Lambda \otimes \Lambda' \lr \Z$. There is also a map $\rot: \Lambda \lr \Lambda'$, that is induced from the map that, up to homotopy, rotates the boundary components clockwise. This induces a bilinear form $\wt{\omega}$ on the first homology $\Lambda$, given by $\wt{\omega}(\gamma, \gamma') = \gamma \cdot \rot(\gamma')$, and we also consider its anti-symmetrization $\omega$. In order to prove the symplecticity of $\omega_{\beta\cdot \Delta^2}$ stated in Theorem <ref>, we use the following result:

There exists a marked surface $(\cS, A, B)$ such that $\Lambda$ is identified with the co-character lattice of $\mathfrak{M}$, and the induced form on the co-character lattice of $\mathfrak{M}$ is identified with $\omega$.

Note that the surface $\cS$ is homeomorphic to the spectral curve constructed in [19], see Section <ref> for details. Now, we need two more properties of $(\cS, A, B)$, which follow from the construction in <cit.>:

- The connected components of $\partial\cS$ correspond to the cycles of $\pi = \pi(\beta)$, i.e. to the components of the closure of the braid $\beta$.

- Let $C$ be the connected component of $\partial\cS$ corresponding to the cycle $(a_{1}\dots a_{k})$. Then, the elements of $A = \{1, \dots, n\}$ appearing in $C$ are precisely $a_{1}, \dots, a_{k}$, and they appear in the same order as in the cycle.

Let us denote by $C_{1}, \dots, C_{k}$ the connected components of $\partial\cS$. The cycle associated to $C_{j}$ will be denoted by $(a_{j, 1}\dots a_{j, \ell_{j}})$. We will now decompose $\Lambda = H_{1}(\cS, A)$, as follows. First, we have the exact sequence in relative homology

0 \to H_{1}(\cS) \to H_{1}(\cS, A)\buildrel \partial \over \to H_{0}(A) = \Z^{A} \to H_{0}(\cS) \to 0,

where the image of $\partial$ is spanned by elements of the form $a - b$, where $a, b \in A$ belong to the same connected component of $\cS$.  For each such $a, b$, we choose a path from $a$ to $b$ in $\cS$, and we let $K$ be the span of the classes of these paths in homology. This gives a splitting

H_{1}(\cS, A) = H_{1}(\cS) \oplus K

We construct a basis of $K$ as follows. For simplicity, we will assume that $\cS$ is connected, the general case follows similarly. For each connected component $C_{j}$ of $\partial\cS$, we take the path from $a_{j, i}$ to $a_{j, i+1}$ following $C_{j}$, $j\in[1,\ell_{j} - 1]$. We also take a path $\gamma_{j}$ from $a_{j, \ell_{j}}$ to $a_{j+1, 1}$, $j\in[1,k-1]$. Then we obtain the basis of $K$, see Figure <ref>:

$$K = \Z\{a_{j,i}a_{j,i+1}, \gamma_{j'} \mid j\in[1,k], i\in[1,\ell_{j} - 1], j'\in[1,k-1]\}.$$

The surface $\cS$, with the marked points in the boundary. Points of $A$ are colored white, and points of $B$ are colored black. For the sake of readability we do not label the paths along the boundary for two consecutive points of $A$.

We can further split $H_{1}(\cS)$ as follows. We let $\bar{\cS}$ be the surface obtained from $\cS$ by attaching disks along the boundary components. We have an exact sequence

0 \to H_{2}(\bar{\cS}) \to H_{1}(\partial \cS) \to H_{1}(\cS) \to H_{1}(\bar{\cS}) \to 0

so that $H_{1}(\cS) = H_{1}(\bar{\cS}) \oplus (H_{1}(\partial \cS)/H_{2}(\bar{\cS}))$. Note that a spanning set for $H_{1}(\partial\cS)/H_{2}(\bar{\cS})$ is given by $C_{i} - C_{j}$, where $C_{i}$ and $C_{j}$ are boundary components of the same connected component of $\cS$. Since we are assuming $\cS$ is connected, a basis is given by $C_{i} - C_{i+1}, i\in[1,k-1]$. Moreover, since the elements in $H_{1}(\cS)$ are $\rot$-invariant, the form on $H_{1}(\cS)$ is given by the intersection form. This implies that $\omega|_{H_{1}(\bar{\cS})}$ is the intersection form on $\bar{\cS}$, and therefore is non-degenerate. In addition, $\omega(H_{1}(\bar{\cS}), H_{1}(\partial \cS)/H_{2}(\bar{\cS})) = 0$ and $\omega(H_{1}(\bar{\cS}), K) = 0$. Thus, using the decomposition

H_{1}(\cS, A) = H_{1}(\bar{\cS}) \oplus (H_{1}(\partial \cS)/H_{2}(\bar{\cS})) \oplus K,

the form $\omega$ has the following form

\omega = \left(\begin{matrix} \omega|_{H_{1}(\bar{\cS})} & 0 & 0 \\ 0 & 0 & * \\ 0 & * & * \end{matrix} \right).

We will not find the remaining terms * for $\omega$, we will only do so after passing to the quotient by the action of a torus, as this is all that suffices. We have a map $\psi: \Z^{A} \to H_{1}(\cS, A)$ that to each point $a \in A$ associates the path that follows the boundary component containing $a$ from $a$ to $\rot^{2}(a)$. In other words, it sends $a_{j, i}$ to a path $a_{j,i}a_{j,i+1}$, where $a_{j, \ell_{j} + 1} = a_{j, 1}$.

Now let $T_{\pi} \subseteq (\C^{*})^{n}/\C^{*}$ be the torus given by the equations $t_{a_{1, \ell_{1}}} = t_{a_{2, \ell_{2}}} = \cdots = t_{a_{k, \ell_{k}}}$. We recall, see Remark <ref> and Corollary <ref>, that $T_{\pi}$ acts freely on the Mellit chart $\mathfrak{M}$. The torus  $T_{\pi}$ is the fixed torus for the action of the element $\sigma = (a_{1,\ell_{1}}a_{2,\ell_{2}}\cdots a_{k, \ell_{k}})$. According to [68], to find the co-character lattice of $\mathfrak{M}/T_{\pi}$ we need to mod out by the image of $\psi$ on $\sigma$-invariant elements of $A$. Thus, the co-character lattice of $\mathfrak{M}/T_{\pi}$ can be identified with

H_{1}(\bar{\cS}) \oplus (H_{1}(\partial\cS)/H_{2}(\bar{\cS})) \oplus \Z\{\gamma_{1}, \dots, \gamma_{k-1}\}.

In addition, note that we can identify $C_{i} = a_{i, \ell_{i}}a_{i,1}$. Thus, $C_{i} - C_{i+1} = a_{i, \ell_{i}}a_{i,1} - a_{i+1, \ell_{i+1}}a_{i+1, 1}$. Note also that $\omega(\gamma_{i}, \gamma_{j}) = 0$ as $\gamma_{i}\cdot \rot(\gamma_{j}) = 0$ for every $i \neq j$. Moreover, $\gamma_{i}\cdot \rot(a_{i, \ell_{i}}a_{i,1} - a_{i+1, \ell_{i+1}}a_{i+1, 1}) = 0$ while $(a_{i, \ell_{i}}a_{i,1} - a_{i+1, \ell_{i+1}}a_{i+1, 1})\cdot \gamma_{i} = 2$. Thus, $\omega(\gamma_{i}, a_{i, \ell_{i}}a_{i,1} - a_{i+1, \ell_{i+1}}a_{i+1, 1}) = 2$. Similarly, we can see that $\omega(\gamma_{i}, a_{i-1, \ell_{i-1}}a_{i-1,1} - a_{i, \ell_{i}}a_{i, 1}) = 1$ and $\omega(\gamma_{i}, a_{i+1, \ell_{i+1}}a_{i+1,1} - a_{i+2, \ell_{i+2}}a_{i+2, 1}) = -1$.  It follows that the form $\omega_{\mathfrak{M}/T_{\pi}}$ is given by the following matrix

\omega_{\mathfrak{M}/T_{\pi}} = \left(\begin{matrix} \omega|_{H_{1}(\bar{\cS})} & 0 & 0 \\ 0 & 0 & -P \\ 0 & P & 0 \end{matrix} \right),

where $P$ is the $(k-1) \times (k-1)$-matrix

P = \left(\begin{matrix} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \cdots &  0 \\ 0 & -1 & 2  & \cdots &  0 \\  \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 2 \end{matrix}\right).

This implies that the 2-form $\omega_{\mathfrak{M}/T_{\pi}}$ is non-degenerate and thus the chart ${\mathfrak{M}/T_{\pi}}$ is (holomorphic) symplectic. Let us now use the above discussion, and this result for the Mellit chart, to conclude Theorem <ref>.(iii).

Let $\beta \in \mathrm{Br}^{+}_{n}$ be a positive braid (word). Then, the 2-form on $\omega_{\beta\cdot\FT}$ induces a 2-form on the augmentation variety $\Aug(\beta, \ft_{c})$ that has maximal rank at every point. Thus, the augmentation variety of any positive braid is holomorphic symplectic.

The augmentation variety $\Aug(\beta, \ft_{c})$ can be identified with $X_{0}(\beta \cdot \Delta; w_{0})/T_{\pi}$, up to the choice of marked points $\ft_{c}$. The coefficients of the form $\omega_{\beta\cdot \FT}$ are regular functions on $X_0(\beta\cdot\FT)$, and it is readily verified that the form is $T$-equivariant. Thus, we have an induced closed $2$-form $\omega_{\beta}$ on the augmentation variety, and it is non-degenerate if and only if its determinant does not vanish anywhere. Let us first prove that it is non-degenerate on all toric charts. Thanks to the discussion above on the Mellit chart, the form $\omega_{\beta}$ is non-degenerate on the (quotient) toric chart $\mathfrak{M}/T_{\pi}$; recall that $\mathfrak{M}$ is the toric chart obtained from opening the crossings in the order given by (the proof of) Theorem <ref>. By Lemma <ref>, the 2-form has constant coefficients in canonical coordinates in any other chart $\mathfrak{M}'/T_{\pi}$ and, by the above, it is non-degenerate on the intersection with $\mathfrak{M}/T_{\pi}$. Thus, the 2-form is non-degenerate on the entire (other) chart $\mathfrak{M}'/T_{\pi}$. Finally, by Theorem <ref>, the toric charts cover $\Aug(\beta, \ft_{c})$ up to codimension 2. Hence, the determinant of $\omega_{\beta}$ is non-zero outside of codimension 2 locus and hence it is non-zero everywhere.

This concludes the proof of Theorem <ref> and establishes that the augmentation variety associated to a positive braid is holomorphic symplectic. This subsection also concludes the first part of the article, and we now move forward to discuss correspondences between braid varieties and the diagrammatic calculus we develop for their study.

§ THE COMBINATORICS OF ALGEBRAIC WEAVES

This section introduces algebraic weaves, a diagrammatic calculus to study the braid varieties $X_0(\beta)$. The present section focuses on the combinatorial aspects of these diagrams; in particular, this formalizes the weave category $\mathfrak{W}_n$ discussed in Section <ref>. We use these algebraic weaves in Section <ref>, where we prove that a weave between two positive braid $\beta_1$ and $\beta_2$ yields a correspondence between the braid varieties $X_0(\beta_1)$ and $X_0(\beta_2)$ (as stated in Theorem <ref>). We refer the reader to [19] for the contact and symplectic geometry motivation behind algebraic weaves.

§.§ Algebraic weaves

Algebraic weaves are planar diagrams introduced in the work of the first author and E. Zaslow [19]. In appearance, these are similar to the planar diagrams appearing in Soergel calculus [30, 31]; there are nevertheless key distinctions, and we refer to our diagrams as weaves, following the symplectic constructions in [19].

A 3-weave from $\beta_2=(\sigma_1\sigma_2)^4\sigma_1\in \Br^+_3$ to $\beta_1=\sigma_2\sigma_1\sigma_2\in Br^+_3$ (Left). The blue color indicates a transposition label $s_1\in S_3$ and the red color indicates the transposition label $s_2\in S_3$. A 2-weave from $\beta_2=\sigma_1^{16}\in \Br_2^+$ to $\beta_1=\sigma_1^{2}\in \Br_2^+$, all black edges are labeled with the unique transposition $s_1\in S^2$.

Let $\beta_1,\beta_2$ be two positive $n$-braid words. A weave of degree $n$ is the image of a continuous map

$$\mathfrak{w}: \bigcup_{i=1}^{n-1} G_i\lr [1,2]\times\R,$$

where each $G_i$, $i\in[1,n-1]$ is a trivalent graph and the following conditions are satisfied:

(i) The restriction $\mathfrak{w}|_{G_i}:G_i\lr[1,2]\times\R$ is a topological embedding for all $i\in[1,n-1]$, which is a smooth embedding away from the trivalent vertices of the graph $G_i$.

(ii) The images $\mathfrak{w}(G_i)$ and $\mathfrak{w}(G_{i+1})$ are only allowed to intersect at trivalent vertices, $i\in[1,n-2]$, and the planar edges around this intersection point must alternatingly belong to $G_i$ and $G_{i+1}$. In addition, intersections between $\mathfrak{w}(G_i)$ and $\mathfrak{w}(G_j)$, $|i-j|\geq2$ are transverse.

(iii) In a neighborhood of $\{j\}\times\R\sse[1,2]\times\R$, $j=1,2$, the image $im(\mathfrak{w})$ is given by $l(\beta_j)$ vertical lines, such that the $k$th line belongs to $G_{\sigma^{(j)}_{i_k}}$, where $\sigma^{(j)}_{i_k}$ is the $k$th crossing of $\beta_j$.$\Box$

The six local models for an algebraic $n$-weave, where $j,k\in[1,n-1]$, except in $(b)$ where $k\in[1,n-2]$, and $|j-k|\geq2$. The inverse of the local model in $(b)$, with $s_{k+1}s_ks_{k+1}$ on top and $s_ks_{k+1}s_k$ on the bottom, is also allowed.

Definition <ref> is precise, but admittedly obtuse; see Figure <ref> for two explicit examples with $n=2,3$. Intuitively, we start with $\beta_2$ being represented by a collection of points in $\{2\}\times\R$, e.g. $(2,k)\in\{2\}\times\R$, $k\in[1,\ell(\beta_2)]$ where we label the $k$th point with the transposition $s_k\in S_{\ell(\beta_2)}$ associated to the $k$th crossing of $\beta_2$. Then we start drawing lines going down, towards $\{1\}\times\R$, from each of the points $(2,k)$, which are the crossings of $\beta_2$. Each of these lines, which we will call edges, are labeled by the same transposition $s_k$ as their starting (highest) point. As we keep drawing the edges downwards, one of the following six situations, depicted in Figure <ref>, might occur:

(a) Two consecutive edges labeled with the same transposition $s_k$ come together, and continue moving down as one unique edge, also labeled with $s_k$, $k\in[1,n-1]$. This is referred to as a trivalent vertex, and correspond to the model around (the image of) a trivalent vertex of the graph $G_k$ in Definition <ref>. Algebraically, we represent this local model by $s_ks_k\to s_k$.

(b) Three consecutive edges labeled by $s_k,s_{k+1},s_k$ come together, and continue moving down as three edges but now labeled $s_{k+1},s_{k},s_{k+1}$. This is referred to as a hexavalent vertex, and correspond to the model around an intersection point of the (images of the) graphs $\mathfrak{w}(G_k)\cap \mathfrak{w}(G_{k+1})$ in Definition <ref>. Algebraically, we represent this local model by $s_ks_{k+1}s_k\to s_{k+1}s_ks_{k+1}$. In addition, we also allow the same move, but reversed: $s_{k+1}s_ks_{k+1}\to s_ks_{k+1}s_k$, with $s_{k+1}s_ks_{k+1}$ on top and $s_ks_{k+1}s_k$ at the bottom.

(c) Two consecutive edges labeled with two different transpositions $s_k,s_j$, with $|j-k|\geq2$, come together, and continue moving down as two edges, now labeled by $s_j,s_k$. This is referred to as a 4-valent vertex, and correspond to the model around a (transverse) intersection point of $\mathfrak{w}(G_k)$ and $\mathfrak{w}(G_j)$ in Definition <ref>. Algebraically, we represent this local model by $s_ks_j\to s_js_k$.

(d) Two consecutive edges labeled with the same transposition $s_k$ come together, merge and there is no edge continuing down. This is referred to as a cup, and we represent this local model by $s_ks_k\to 1$.

(e) The inverse of the move in (d), where two consecutive edges are created as moving downwards from the empty set. This is referred to as a cap, and we represent this local model by $1\to s_ks_k$.

(f) There is an edge labeled by $s_k$ and it continues moving down as the same edge labeled by $s_k$, i.e. nothing occurs. This local model is represented algebraically by $s_k\to s_k$.

Local models for weaves with yellow segments.

In the six models (a)-(f) above, the algebraic representation is merely the result of taking horizontal cross-sections along the weave (or local model) and reading the permutations $s_k$ (or braid group generators $\sigma_k$), left to right, that label the edges intersecting the cross-section. For instance, four-valent and hexavalent vertices represent (the Coxeter projection of the) braid relations. In the course of the manuscript, we will also use yellow segments in order to keep track of certain variables, corresponding to the $z_i$-variables in the braid variety. In addition, we also consider particular types of weaves. This is the content of the following:

An algebraic weave is a weave such that:

- The edges have been oriented downwards, with the models according to Figure <ref> for cups and caps. By convention, diagrams are oriented from top to bottom, from $\{2\}\times\R$ down to $\{1\}\times\R$.

- Trivalent vertices, cups and caps are decorated with yellow segments to the left.[These yellow segments are not considered part of the graph, and in particular they are not edges, i.e. a trivalent vertex is still considered trivalent even if there is a yellow segment from it.] See Figure <ref> for a depiction of these yellow segments. Near another vertex, we allow a yellow line to intersect either all incoming or all outgoing edges.

The following two are special types of weaves that we will use:

(ii) A simplifying weave is an algebraic weave with no caps; thus the only allowed local models are (a),(b),(c),(d) and (f), not (e).

(ii) A Demazure weave is an algebraic weave with no cups nor caps; thus the only allowed local models are (a),(b),(c) and (f), not (d),(e).

It is readily seen that an algebraic weave is simplifying if and only if the length of a braid word is not increasing as we scan down the weave with horizontal cross-sections. The reasons behind the choice of the name “Demazure weaves” will be explained in Section <ref>.

Note that upside-down trivalent vertices, i.e. the horizontal flip of model (a), given by $s_k\to s_ks_k$, can be constructed using the above trivalent vertices and caps. In fact, an upside-down trivalent vertex can be created in at least the following ways:

In the Subsection <ref>, these different ways of drawing upside-down vertices are declared to be equivalent. Finally, an (algebraic) $n$-weave $\Sigma$ will almost always be considered through its horizontal cross-sections, and thus we will typically refer to it as being a sequence of (consecutively distinct) positive braid words
$$\beta_j(\Sigma)=s^{(j)}_{i_1}s^{(j)}_{i_2}\cdots s^{(j)}_{i_{\ell(\beta_j)}}\in \Br_n^+,\quad j\in [0,\ell(\Sigma)],$$

where the only changes are the ones specified in the local models (a)-(f) above. In this case, we use the notation

$$\Sigma=\beta_0(\Sigma)\to\beta_1(\Sigma)\to\cdots\to \beta_{\ell(\Sigma)}(\Sigma).$$

The initial braid word $\beta_0(\Sigma)$ being read at the horizontal cross-section $\{2\}\times\R$, and the last braid word $\beta_{\ell(\Sigma)}(\Sigma)$ is read at the horizontal cross-section $\{1\}\times\R$. The number $\ell(\Sigma)\in\N$ will be referred to as the length of the weave $\Sigma$, the initial braid word $\beta_0(\Sigma)$ as the convex end of the weave, and the last braid word $\beta_{\ell(\Sigma)}(\Sigma)$ as the concave end of the weave. The convexity and concavity notation aims at agreeing with the contact topological framework developed in [19].

§.§ Equivalence of weaves
In this section, we impose certain equivalences between two algebraic weaves $\Sigma_1,  \Sigma_2$ whose convex and concave ends coincide, i.e. $\beta_0(\Sigma_1)=\beta_0(\Sigma_2)$ and $\beta_{\ell(\Sigma_1)}(\Sigma_1)=\beta_{\ell(\Sigma_2)}(\Sigma_2)$. These equivalences $\Sigma_1\simeq \Sigma_2$ are the result of imposing a Hamiltonian isotopy equivalence to the (exact) Lagrangian surfaces obtained by projecting to $(T^*([1,2]\times\R^2),d\la_\st)$ the Legendrian lifts of these weaves $\Sigma_1, \Sigma_2$ in the standard contact 5-dimensional Darboux chart $(J^1([1,2]\times\R^2),\xi_\st)$. The geometric proofs are explained in detail in <cit.>. In the present article, we directly write the diagrammatic equivalences, which can be understood as moves between braid isotopies (also known as movies moves).

These equivalences are chosen as dictated by contact topology but can be now taken to be diagrammatic. The remarkable algebraic properties of these equivalences that we list below is that the braid matrices that we will associate to the weaves in Section <ref> will in fact preserve these equivalences. In particular, an algebraic reason not to declare the mutated weaves equivalent is that they will not preserve these braid matrix relations.$\Box$

Each of the following items represents an equivalence between weaves. The majority of equivalences we describe compare two local models, and an equivalence between two different weaves $\Sigma_1, \Sigma_2$ will be obtained by applying several of the local equivalences listed here.

§.§.§ Planar isotopies
A weave is, in particular, a planar diagram: we declare planar isotopic diagrams to define equivalent weaves. As a first consequence, the two ways to declare an upside down trivalent vertex are the same, and we have the following additional relations:

These planar isotopies do include canceling pairs of caps and cup, as depicted above. For instance, the weave $\Sigma_1=s_k\to s_ks_ks_k\to s_k\cdot 1=s_k$, where first a cap creates $1\to(s_ks_k)$ to the left of the initial $s_k$, and then a cup erases the rightmost to $s_k$ via $(s_ks_k)\to1$, is equivalent to the constant weave $\Sigma_2=s_k$.

§.§.§ Canceling pairs of 4- and 6-valent vertices
The following weaves are declared to be equivalent:

From the algebraic perspective, i.e. studying the braids in the horizontal cross-sections, this the diagrammatic incarnation of the fact that the two moves $s_ks_{k+1}s_k\to s_{k+1}s_ks_{k+1}$ and $s_{k+1}s_ks_{k+1}\to s_ks_{k+1}s_k$, and the two moves $s_is_j\to s_js_i$ and $s_js_i\to s_is_j$, $|i-j|\geq2$, are inverse to each other. That is, performing a Reidemeister III move and then its inverse is considered to be (equivalent to) the trivial weave. Similarly, performing a commutation move in the braid group, and then the same move in reverse, is also considered to be (equivalent to) the trivial weave. In the notation above, we are declaring the weave $\Sigma_1=s_{k+1}s_ks_{k+1}\to s_ks_{k+1}s_k\to s_{k+1}s_ks_{k+1}$ to be equivalent to the constant weave $\Sigma_2=s_{k+1}s_ks_{k+1}$, and the weave $\Sigma_1=s_{i}s_j\to s_js_i\to s_is_j$ to be equivalent to the constant weave $\Sigma_2=s_is_j$.

§.§.§ Commutation with distant colors
We declare that an edge of the weave labeled with a color (i.e. a transposition) which is distant to the rest of the colors at a given vertex can be moved past this vertex. That is, we declare that the following weaves are equivalent:

Similarly, we declare that three lines with pairwise distant colors can be rearranged according to the weave equivalence depicted above. As illustrated in the equivalences above, the particular sequences of braid moves that we are declaring to be equivalent are readily read from taking horizontal cross-sections in the above diagrams; we will thus not indicate them any longer.

§.§.§ 1212- and 2121-relations

We require that the following two ways of getting from $\sigma_1\sigma_2\sigma_1\sigma_2$, denoted 1212 for simplicity, to $\sigma_1\sigma_2\sigma_1$, i.e. 121, are equivalent:

Note that we interpret the same pentagon, on the left of the above figure, in several different ways. Namely, this requirement also means that the two ways of getting from 1121 to 212 are equivalent, and that the two ways of getting from 2212 to 121 are equivalent, and so forth. The equivalence of the two ways of getting from 1121 to 212 corresponds to the equivalence of the following two simplifying weaves:

We also require that the two ways of getting from 1211 to 212 are equivalent, which is the same as requiring that the two ways of getting from 2121 to 212 are equivalent and so on. The weaves are obtained from the ones above by the symmetry along the vertical line:

There are also similar relations where adjacent colors are interchanged (e.g. black and red), and for any pair of adjacent colors, which we do not draw them here.

§.§.§ Cycles for 12121

As an example for the previous relation, we observe that there are many paths in the Demazure graph from 12121 to 212, related by consecutive application of the 1212-relation:

(A)\ 12121\to 21221\to 2121\to 2212\to 212

(B)\ 12121\to 11211\to 1211\to 2121\to 2112\to 212

(C)\ 12121\to 11211\to 1211\to 121\to 212 \sim 12121\to 11211\to 1121\to 121\to 212

(D)\ 12121\to 11211\to 1121\to 1212\to 2122\to 212

(E)\ 12121\to 12212\to 1212\to 2122\to 212

§.§.§ Zamolodchikov relation

Diagrammatically, the Zamolodchikov relation is the equivalence of the following diagrams, relating various braid words for the longest element $w_0\in S_4$:

§.§.§ More moves with cups

By rotating the above relations, we obtain many interesting relations with cups and caps. We include just a few here:

Note that these two pictures, as planar graphs, are similar to the ones that we already considered (cancellation $121\to 212\to 121$ and $1212$-move), but in this case they are drawn differently. The following relations corresponds to different ways to a look at a single 6-valent vertex:

§.§.§ Mutations

In contrast with Soergel calculus, we do not declare two ways of getting from $s_is_is_i$ to $s_i$ via $s_is_i$ to be equivalent. They are related by the following special type of move, which we call a weave mutation:

This concludes the list of diagrammatic equivalences (<ref>)-(<ref>), and the mutation non-equivalence (<ref>).

§.§ Demazure product and Demazure weaves

The NilHecke algebra $NH_n$  is defined by generators $e_1,\ldots,e_{n-1}$ and relations $e_i^2=e_i$ and

e_{i}e_{i+1}e_{i}=e_{i+1}e_{i}e_{i+1},\, \, \, \, \ e_ie_j=e_je_i, \, \ |i-j|\ge 2.

We have a natural homomorphism from the braid group to $NH_n$ which we will denote by $\delta$ following [60].
We will denote by $e_w$ the image of the positive reduced braid lift of $w\in S_n$ under $\delta$. It is clear that for any braid $\beta$ there exists $w$ such that $\delta(\beta)=e_w$. By abuse of notation, we will write $\delta(\beta)=w$ and call $w$ the Demazure product of $\beta$. This terminology is used by Knutson and Miller [60], but the notion goes back at least to Demazure [26].
Given two permutations $u,v\in S_n$, we define $u\star v=\delta(uv)$. By construction, we have

(u\star v)\star w=\delta(uvw)=u\star (v\star w).

We will call the star product of the Coxeter generators corresponding to the letters in a positive braid word the Demazure product of the word. For a braid word $\sigma_{i_1} \sigma_{i_1} \ldots \sigma_{i_l(\beta)}$ for a positive braid $\beta,$ we have

e_{i_1} \star e_{i_2} \star \ldots \star e_{i_l(\beta)} = \delta(\beta).

In other words, the Demazure product of a braid word for $\beta$ equals $\delta(\beta).$ In particular, it does not depend on the choice of a word for $\beta$.

It is readily verified that $w\star s_i=ws_i$ if $\ell(ws_i)=\ell(w)+1$ and $w\star s_i=w$ if $\ell(ws_i)=\ell(w)-1$. The Demazure power is simply $s_i \star s_i \star \ldots \star s_i = s_i,$ for any number of multiples, and we also have the equality $w\star w_0=w_0$.$\Box$

The NilHecke algebra $NH_n$ is the monoid algebra of the monoid defined by the same generators and relations, whose multiplication is given by the Demazure product. For this reason, the latter is also known as the 0-Hecke product; the monoid is called Coxeter monoid [90], $0-$Hecke monoid [34, 48], or Richardson-Springer monoid by different authors. Richardson and Springer studied its action on the set of orbits of the flag variety under the action of the fixed point subgroup of an involution on the algebraic group [81, 82]. The elements of this monoid are precisely $\left\{e_w | w \in S_n\right\}$, i.e. we have a bijection between the set $S_n$ and the underlying set of the monoid (this was proved by Norton [73]). As the examples above show, the multiplication in the monoid is quite different from the one in the permutation group. More generally, given a positive braid $\beta$, $\delta(\beta)$ does not coincide with the image of $\beta$ under the canonical surjection onto $S_n$. A first relation to the weaves introduced above is given in the following simple:

Let $\Sigma$ be a Demazure weave. Then the Demazure product of the associated braid words $\beta_j(\Sigma)$, $j\in[0,l
(\Sigma)]$, remains unchanged, i.e. $\delta(\beta_0(\Sigma))=\delta(\beta_j(\Sigma))$.

By the arguments above, the Demazure product is invariant under braid moves in a word, so $6$- and $4$-valent vertices do not change it. Example <ref> shows that $3$-valent vertices also preserve the Demazure product.

Lemma <ref> shows that Demazure weaves provide a transparent diagrammatic interpretation of the Demazure product and of the $0-$Hecke monoid. This motivated our nomenclature.

§.§ Classification of algebraic weaves

We call two algebraic weaves equivalent if they are related by a sequence of elementary equivalence moves from the previous section (with no mutations), and mutation equivalent if they are related by a sequence of equivalence moves and mutations.

$($a$)$ Let $\Sigma_1,\Sigma_2$ be two algebraic weaves with the same source and target braids. If $\Sigma_1,\Sigma_2$ only have 6- and 4- valent vertices, then $\Sigma_1,\Sigma_2$ are equivalent.

$($b$)$ Let $\Sigma_1,\Sigma_2$ be two Demazure algebraic weaves with the same source and target. If the target is reduced, then $\Sigma_1,\Sigma_2$ are mutation equivalent.

The theorem follows from the main result of [29], which we briefly recall. For part (a), consider the graph where vertices correspond to braid words and edges to braid moves (that is, 6- or 4-valent vertices). Then the cycles in this graph are generated by commutation with distant colors and Zamolodchikov relations, hence any two paths in this graph are equivalent.

For (b), consider the Hecke-type algebra with generators $T_i$ and relations

T_i^2=\alpha T_i+\beta,\ T_iT_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}+\text{lower order terms},\ T_iT_j=T_jT_i, (|i-j|>1).

Using these relations, it is easy to see that every product of $T_i$ can be written as a linear combination of reduced expressions, possibly in a non-unique way. This non-uniqueness appears from ambiguities: applying the relations in different order could yield a different results.

B. Elias proved in <cit.> that (modulo commutation with distant colors) there are exactly 5 types of potential abmiguities that one needs to consider: $iii,ii(i+1)i, i(i+1)ii, i(i+1)i(i+1)i,i(i+1)(i+2)i(i+1)i$, which are nothing but the trivial move, the 5-cycles corresponding to 1121 and 1211 from <ref>, the cycle from  <ref> for the word 12121, and the Zamolodchikov relation. Note that the ambiguity $iii$ corresponds to different ways of getting from $iii$ to $i$. There are two such ways without cups, and they are related by the first mutation from Section  <ref>.

The assumption in (b) that the target is reduced is important. Indeed, the two trivalent vertices $(ss)s\to ss$ and $s(ss)\to ss$ are not mutation equivalent.

Note that by Theorem <ref>(a), any two simplifying weaves relating two braid words for the same braid are equivalent. Thus, we will not specify such a weave.

Let us continue studying conditions for equivalences, it is indeed useful to have different criteria when verifying that two weaves are equivalent (or mutation equivalent). Suppose that a braid word $\beta$ contains a piece $s_ius_j$. Following [47], the pair of crossings $(s_i,s_j)$ is said to be a deletion pair if $s_iu=us_j$.[Note that unlike [47], we do not require $u$ to be a reduced word.] Let us define a relation $\prec$ on the crossings of the braid $\beta$ according to $s_{i} \prec s_{j}$ if $(s_{i}, s_{j})$ form a deletion pair. The following two lemmas are used in the proof of the criterion Theorem <ref> below:

(i) The relation $\prec$ is a partial order on the set of crossings of $\beta$.

(ii) The set of crossings of $\beta$ is a disjoint union of linearly ordered sets.

Assume that we have a piece of a braid $s_ius_jvs_k$ and $(s_i,s_j)$ and $(s_j,s_k)$ are deletion pairs so that $s_iu=us_j, s_jv=vs_k$. Then

s_i\cdot us_jv=us_js_jv=us_jv\cdot s_k,

and $(s_i,s_k)$ is a deletion pair. This proves (i). To prove (ii), assume  $(s_i,s_j)$ and $(s_i,s_k)$ are deletion pairs, and assume wlog that $s_{j}$ is to the left of $s_{k}$. We must show that $(s_{j}, s_{k})$ is a deletion pair. We have

s_iu=us_j,\ s_ius_jv=us_jvs_k

and $s_ius_jv=s_iuvs_k$, hence $s_jv=vs_k$ and $(s_j,s_k)$ is a deletion pair. The case when $(s_i,s_k)$ and $(s_j,s_k)$ are deletion pairs is analogous.

We call a deletion pair $(s_i,s_j)$ close if $s_{i} \prec s_{j}$ is a cover relation, i.e. no crossing in-between $s_{i}$ and $s_{j}$ forms a deletion pair with $s_{i}$ or $s_{j}$.

Suppose that $(s_i,s_j)$ is a close deletion pair, then the following Demazure weaves are equivalent:

s_ius_j\to s_is_iu\to s_iu\to us_j\ \sim \ s_ius_j\to us_js_j\to us_j.

Note that the condition that the deletion pair is close is necessary, see Remark <ref>.

We prove the statement by induction by the length of $u$. If $u$ is empty, the statement is clear. Otherwise, by definition of deletion pair we get $s_iu=us_j$. If $u$ ends with $s_j$ then we do not have a close pair, contradiction. Otherwise we need to apply some braid relation to $us_j$ which involves $s_j$. We have the following cases:

1) If $u=vs_k$ and $|k-j|>1$ then $us_j=vs_ks_j=vs_js_k$ while $s_iu=s_ivs_k$, so $vs_j=s_iv$. By the assumption of induction, two Demazure weaves corresponding to $s_ivs_js_k$ are equivalent, and we get the following diagram:

s_ivs_ks_j r d    vs_js_ks_j r d    d vs_ks_js_j

s_ivs_js_k r d    vs_js_js_k dr    vs_ks_j

s_is_ivs_k r    s_ivs_k r    vs_js_k u

The top square is an isotopy, and the pentagon on the right is commutation with distant colors.

2) If $u=vs_js_{j+1}$ then $us_j=vs_js_{j+1}s_j=vs_{j+1}s_js_{j+1}$ while $s_iu=s_ivs_js_{j+1}$, so $s_iv=vs_{j+1}$. By the assumption of induction, two Demazure weaves corresponding to $s_ivs_{j+1}$ are equivalent, and we get the following diagram:

s_ivs_js_j+1s_j r d    vs_j+1s_js_j+1s_j r d    d vs_js_j+1s_js_j

s_ivs_j+1s_js_j+1 r d    vs_j+1s_j+1s_js_j+1 dr    vs_js_j+1s_j

s_is_ivs_js_j+1 r    s_ivs_js_j+1 r    vs_j+1s_js_j+1 u

The top square is an isotopy, and the pentagon on the right is 5-cycle from Section <ref>. The case when $u=vs_js_{j-1}$ is analogous.

Given a Demazure weave $\Sigma$, we have an injection $\iota_{\Sigma}$ from the set of crossings in the target to the set of crossings in the top: for a 6-valent vertex  it is a bijection which exchanges left and right crossings, for a 4-valent vertex it is a bijection exchanging crossings, and for a 3-valent vertex the injection sends the crossing in the target to the right crossing in the source. We refer to the crossings not in the image of $\iota_{\Sigma}$ as missing. The following result is a characterization of the equivalence between Demazure weaves:

Suppose that $\ell(\beta_1)=\ell(\beta_0)-1$. Then two Demazure weaves from $\beta_0$ to $\beta_1$ are equivalent if and only if they have the same missing crossing in $\beta_0$.

Since $\ell(\beta_1)=\ell(\beta_0)-1$, any Demazure weave between $\beta_0$ and $\beta_1$ has one trivalent vertex. Let us prove that equivalent weaves have the same missing crossing. It is easy to see that commutations with distant colors and Zamolodchikov relations induce the same bijections between crossings, so any two weaves with the same source and target and only 6- and 4-valent vertices induce the same bijection. Finally, for the 5-cycle from Section <ref> we observe that in either weave for 1121 the first crossing is missing, while in either weave for 1211 the third crossing is missing.

Conversely, assume that the Demazure weaves $\Sigma_{1}, \Sigma_{2}: \beta_0 \to \beta_1$ have the same missing crossing. It is sufficient to prove that they can be related by a sequence of cycles from Lemma <ref> and equivalences. Note that a trivalent vertex corresponds to a close deletion pair. We have the following cases, where the deletion pair is underlined:

1) Assume that $(s_i,s_j)$ is a close deletion pair and we apply a 4-valent vertex to $s_j$:

\underline{s_i}u\underline{s_j}s_k=\underline{s_i}us_k\underline{s_j},\ |k-j|>1,

then $(s_i,s_j)$ is a close deletion pair in the resulting braid, and Lemma <ref> applies.

2) Assume that we apply a 6-valent vertex with $s_j$ on the left:

\underline{s_i}u\underline{s_j}s_{j+1}s_{j}=\underline{s_i}us_{j+1}s_j\underline{s_{j+1}}

then $(s_i,s_{j+1})$ is a close deletion pair in the resulting braid, and Lemma <ref> applies.

3) Assume that we apply a 6-valent vertex with $s_j$ on the right:

\underline{s_i}us_js_{j+1}\underline{s_{j}}=\underline{s_i}u\underline{s_{j+1}}s_js_{j+1}

Note that $s_ius_js_{j+1}=us_{j}s_{j+1}s_j=us_{j+1}s_js_{j+1}$ implies $s_iu=us_{j+1}$, and  $(s_i,s_{j+1})$ is again a close deletion pair.

4) Finally, assume that we apply a 6-valent vertex with $s_j$ in the middle, then we no longer get a deletion pair. Instead, $u=vs_{j+1}$ and $s_ivs_{j+1}=vs_{j+1}s_{j}$. By considering possible braid moves, we can write $v=ws_j$, then

hence $s_iw=ws_{j+1}$.We get the following diagram:

s_iws_js_j+1s_js_j+1 r d dr    s_iws_js_js_j+1s_j r     dl ws_j+1s_js_js_j+1s_j

s_iws_j+1s_js_j+1s_j+1  d    ws_j+1s_js_j+1s_js_j+1 r dl    ws_js_j+1s_js_js_j+1d

ws_j+1s_j+1s_js_j+1s_j+1r    ws_j+1s_js_j+1s_j+1 r    ws_js_j+1s_js_j+1