# Stated skein algebras and their representations

Julien Korinman  Department of Mathematics, Faculty of Science and
Engineering, Waseda University,3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555,
Japan

###### Abstract.

This is a survey on stated skein algebras and their representations.

###### Key words and phrases:

Stated skein algebras, quantum Teichmüller spaces, character varieties

###### 1991 Mathematics Subject Classification:

$57$R$56$, $57$N$10$, $57$M$25$.

###### Contents

  1. 1 Introduction
  2. 2 Stated skein algebras
    1. 2.1 Marked surfaces vs punctured surfaces
    2. 2.2 Definition and first properties of stated skein algebras
    3. 2.3 The gluing/cutting/excision formula and triangular decomposition
    4. 2.4 The quantum fusion operation
    5. 2.5 The triangular strategy
    6. 2.6 Chebyshev-Frobenius morphisms
    7. 2.7 Center and PI dimension of stated skein algebras at roots of unity
    8. 2.8 Finite presentations
  3. 3 Quantum Teichmüller theory
    1. 3.1 Quantum tori
    2. 3.2 The balanced Chekhov-Fock algebra
    3. 3.3 Bonahon-Wong’s quantum trace
    4. 3.4 Lê-Yu’s enhancement of the quantum trace
    5. 3.5 Closed surfaces
  4. 4 Relative character varieties
    1. 4.1 Deformation quantization
    2. 4.2 First examples and the work of Semenov-Tian-Shansky
    3. 4.3 The classical fusion operation
    4. 4.4 Relative character varieties
    5. 4.5 Comparison between stated skein algebras and relative character varieties
  5. 5 Representations of stated skein algebras
    1. 5.1 Representations coming from modular TQFTs
    2. 5.2 Representations coming from non semi-simple TQFTs
    3. 5.3 Representations coming from quantum tori
    4. 5.4 The unicity representations theorem and Azumaya locus
    5. 5.5 Brown and Gordon’s Poisson orders
    6. 5.6 Resolutions of the classification problem is small cases
  6. 6 Generalizations to other gauge groups

##  1\. Introduction

Let $\Sigma$ be a compact oriented surface, $k$ be a (unital, associative)
commutative ring and $A\in k^{\times}$ an invertible element. The Kauffman-
bracket skein algebra $\mathcal{S}_{A}(\Sigma)$ is the quotient of the $k$
module freely generated by embedded framed links $L\subset\Sigma\times(0,1)$
by the ideal generated by elements $L-L^{\prime}$, for $L,L^{\prime}$ two
isotopic links, and by the following skein relations:

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The product of two classes links $[L_{1}]$ and $[L_{2}]$ is defined by
isotoping $L_{1}$ and $L_{2}$ in $\Sigma\times(1/2,1)$ and
$\Sigma\times(0,1/2)$ respectively and then setting
$[L_{1}]\cdot[L_{2}]:=[L_{1}\cup L_{2}]$.

Skein algebras have been introduced by Przytycki [Prz99] and Turaev [Tur88] at
the end of the 80’s as a tool to study the $\mathrm{SU}(2)$ Witten-
Reshetikhin-Turaev topological quantum field theories ([Wit89, RT91]). Skein
algebras appear in TQFTs through their finite dimensional representations.
Such representations exist if and only if the parameter $A$ is a root of
unity. Despite the apparent simplicity of their definition, these algebras and
their representations are quite hard to study. A major breakthrough was made
by Bonahon and Wong ([BW11, BW16a]) who introduced three key original concepts
to understand skein algebras which are:

  1. (1)

the stated skein algebras,

  2. (2)

the quantum traces and

  3. (3)

the Chebyshev-Frobenius morphisms.

Since the introduction of these new ideas, the amount of papers on the subject
has grown very fast and many problems which were out of reach recently are now
affordable. The goal of the present survey is to provide a self-content
introduction on these three concepts and their use including proofs when the
latters are short and enlightening. In addition to review some recent results
on the subject, we dress a list of open problems/questions. Our leading
problem will be

######  Problem 1.1.

Classify all finite dimensional weight representations of (stated) skein
algebras when $A$ is a root of unity of odd order.

As we shall see, this problem is deeply connected to the study of the Poisson
geometry of relative character varieties, more precisely to the computation of
their symplectic leaves. Here we choose the order of $A$ to be odd and
restrict to weight representations for simplicity. Actually, even for the
bigon, one of the simplest marked surface, Problem 1.1 is undecidable so we
will reformulate later a more reasonable version in Problem 5.4.

The paper is organised as follows. We first define stated skein algebras and
review their fundamental properties such as their behaviour for the gluing and
fusion operations, their triangular decomposition, their Chebyshev-Frobenius
morphisms and their finite presentations. We then review Bonahon-Wong’s
quantum trace which permits to embed skein algebras into some quantum tori.
The third section reviews the Poisson geometry of relative character
varieties, in particular the classification of their symplectic leaves. In the
last section we review three families of representations of skein algebras
coming from modular TQFTs, non semi-simple TQFTs and quantum Teichmüller
theory. We then present two important theorems concerning the representation
theory of skein algebras which are the Frohman-Lê-Kania-Bartoszynska Unicity
representation theorem and Brown-Gordon’s theory of Poisson orders. Putting
everything together, we’ll review some recent results of Ganev-Jordan-Safranov
and solve Problem 1.1 in some simple cases.

#### Acknowledgments.

This manuscript is an expanded version of a proceeding of the conference
”Intelligence in low dimensional topology” held at the RIMS. The author warmly
thanks T.Ohtsuki for inviting him at the conference and S.Baseilhac,
F.Bonahon, F.Costantino, L.Funar , T.Q.T. Lê, J.Marché, J.Murakami, A.Quesney,
P.Roche and R.Santharoubane for useful discussions on the subject. He
acknowledges support from the Japanese Society for Promotion of Science (JSPS)
and the Centre National de la Recherche Scientifique (CNRS).

##  2\. Stated skein algebras

###  2.1. Marked surfaces vs punctured surfaces

The key concept behind stated skein algebras is the introduction of marked
surfaces.

######  Definition 2.1.

A marked surface $\mathbf{\Sigma}=(\Sigma,\mathcal{A})$ is a compact oriented
surface $\Sigma$ (possibly with boundary) with a finite set
$\mathcal{A}=\\{a_{i}\\}_{i}$ of orientation-preserving immersions
$a_{i}:[0,1]\hookrightarrow\partial\Sigma$, named boundary arcs, whose
restrictions to $(0,1)$ are embeddings and whose interiors are pairwise
disjoint.

An embedding $f:(\Sigma,\mathcal{A})\to(\Sigma^{\prime},\mathcal{A}^{\prime})$
of marked surfaces is a orientation-preserving proper embedding
$f:\Sigma\to\Sigma^{\prime}$ so that for each boundary arc $a\in\mathcal{A}$
there exists $a^{\prime}\in\mathcal{A}$ such that $f\circ a$ is the
restriction of $a^{\prime}$ to some subinterval of $[0,1]$. When several
boundary arcs $a_{1},\ldots,a_{n}$ in $\mathbf{\Sigma}$ are mapped to the same
boundary arc $b$ of $\mathbf{\Sigma}^{\prime}$ we include in the definition of
$f$ the datum of a total ordering of $\\{a_{1},\ldots,a_{n}\\}$. Marked
surfaces with embeddings form a category $\mathrm{MS}$ with monoidal structure
given by disjoint union.

By abuse of notations, we will often denote by the same letter the embedding
$a_{i}$ and its image $a_{i}((0,1))\subset\partial\Sigma$ and both call them
boundary arcs. We will also abusively identify $\mathcal{A}$ with the disjoint
union $\bigsqcup_{i}a_{i}((0,1))\subset\partial\Sigma$ of open intervals. The
main interest in considering marked surfaces is that they have a natural
gluing operation. Let $\mathbf{\Sigma}=(\Sigma,\mathcal{A})$ be a marked
surface and $a,b\in\mathcal{A}$ two boundary arcs. Set
$\Sigma_{a\\#b}:={\raisebox{1.99997pt}{$\Sigma$}\left/\raisebox{-1.99997pt}{$a(t)\sim
b(1-t)$}\right.}$ and $\mathcal{A}_{a\\#b}:=\mathcal{A}\setminus a\cup b$. The
marked surface $\mathbf{\Sigma}_{a\\#b}=(\Sigma_{a\\#b},\mathcal{A}_{a\\#b})$
is said obtained from $\mathbf{\Sigma}$ by gluing $a$ and $b$. We say that
$\mathbf{\Sigma}=(\mathbf{\Sigma},\mathcal{A})$ is unmarked if
$\mathcal{A}=\emptyset$.

A related concept is the notion of punctured surfaces: a punctured surface is
a pair $(S,\mathcal{P})$ where $S$ is a compact oriented surface and
$\mathcal{P}\subset\Sigma$ a finite subset of punctures which non trivially
intersects each connected component of $\partial\Sigma$. To a punctured
surface, one associates a marked surface $(\Sigma,\mathcal{A})$ where $\Sigma$
is obtained from $S$ by blowing up each inner puncture and the boundary arcs
in $\mathcal{A}$ are the connected components of $\partial
S\setminus\mathcal{P}$. Both notions marked surfaces and punctured surfaces
are used in the literature and are essentially equivalent. On the one hand,
marked surfaces have the advantage of having a natural notion of morphisms,
which is hard to translate in the language of punctured surfaces, and
generalize naturally to $3$-manifolds. On the other hand, punctured surfaces
make much more natural the concept of triangulation. In this survey, we will
use marked surfaces $\mathbf{\Sigma}=(\Sigma,\mathcal{A})$ but will call
punctures the connected components of $\partial\Sigma\setminus\mathcal{A}$ and
inner punctures the unmarked connected components of $\partial\Sigma$ (so they
are circles really).

######  Notations 2.2.

Let us name some marked surfaces. Let $D(n)$ be a disc $\mathbb{D}^{2}$ with
$n$ pairwise disjoint open subdiscs removed and let $\Sigma_{g,n}$ be an
oriented connected surface of genus $g$ with $n$ boundary components. We will
sometimes write $\Sigma_{g}=\Sigma_{g,0}$.

  1. (1)

The $n^{th}$-punctured monogon $\mathbf{m}_{n}=(D(n),\\{a\\})$ is $D(n)$ with
one boundary arc in $\partial\mathbb{D}^{2}$.

  2. (2)

The $n^{th}$-punctured bigon $\mathbb{D}_{n}=(D(n),\\{a,b\\})$ is $D(n)$ with
two boundary arcs in $\partial\mathbb{D}^{2}$. We call
$\mathbb{B}:=\mathbb{D}_{0}$ simply the bigon.

  3. (3)

A triangle $\mathbb{T}=(\mathbb{D}^{2},\\{a_{1},a_{2},a_{3}\\})$ is a disc
with three boundary arcs on its boundary. Its boundary arcs are called edges.

  4. (4)

We denote by $\mathbf{\Sigma}_{g,n}^{0}=(\Sigma_{g,n+1},\\{a\\})$ the surface
$\Sigma_{g,n}$ with a single boundary arc in one of its only boundary
component and by
$\underline{\mathbf{\Sigma}}_{g,n}=(\Sigma_{g,n},\mathcal{A})$ the surface
$\Sigma_{g,n}$ with exactly one boundary arc in each of its boundary component
(so $\mathbf{\Sigma}_{g,0}^{0}=\underline{\mathbf{\Sigma}}_{g,1}$).

######  Definition 2.3.

A marked surface $\mathbf{\Sigma}$ is triangulable if it can be obtained from
a finite disjoint union $\mathbf{\Sigma}_{\Delta}=\bigsqcup_{i}\mathbb{T}_{i}$
of triangles by gluing some pairs of edges. A triangulation $\Delta$ is then
the data of the disjoint union $\mathbf{\Sigma}_{\Delta}$ together with the
set of glued pair of edges.

The connected components $\mathbb{T}_{i}$ of $\mathbf{\Sigma}_{\Delta}$ are
called faces and their set is denoted $F(\Delta)$. The image in $\Sigma$ of
the edges of the faces $\mathbb{T}_{i}$ are called edges of $\Delta$ and their
set is denoted $\mathcal{E}(\Delta)$. Note that each boundary arc is an edge
in $\mathcal{E}(\Delta)$; the elements of the complementary
$\mathring{\mathcal{E}}(\Delta):=\mathcal{E}(\Delta)\setminus\mathcal{A}$ are
called inner edges. The punctures of $\mathbf{\Sigma}$ are the vertices of the
triangulation; in particular this set is non empty so $\Sigma$ cannot be
closed.

Note that the only connected marked surfaces which are not triangulable are:
the bigon $\mathbb{B}$, the monogon $\mathbf{m}_{0}$ and the unmarked surfaces
$\Sigma_{g,0}$, $\Sigma_{0,1}$ and $\Sigma_{0,2}$.

###  2.2. Definition and first properties of stated skein algebras

A tangle is a compact framed, properly embedded $1$-dimensional manifold
$T\subset\Sigma\times(0,1)$ such that for every point of $\partial
T\subset\mathcal{A}\times(0,1)$ the framing is parallel to the $(0,1)$ factor
and points to the direction of $1$. The height of $(v,h)\in\Sigma\times(0,1)$
is $h$. If $a$ is a boundary arc and $T$ a tangle, we impose that no two
points in $\partial_{a}T:=\partial T\cap a\times(0,1)$ have the same heights,
hence the set $\partial_{a}T$ is totally ordered by the heights. Two tangles
are isotopic if they are isotopic through the class of tangles that preserve
the boundary height orders. By convention, the empty set is a tangle only
isotopic to itself.

Let $\pi:\Sigma\times(0,1)\rightarrow\Sigma$ be the projection with
$\pi(v,h)=v$. A tangle $T$ is in generic position if for each of its points,
the framing is parallel to the $(0,1)$ factor and points in the direction of
$1$ and is such that $\pi_{|T}:T\rightarrow\Sigma$ is an immersion with at
most transversal double points in the interior of $\Sigma$. Every tangle is
isotopic to a tangle in generic position. We call diagram the image $D=\pi(T)$
of a tangle in generic position, together with the over/undercrossing
information at each double point. An isotopy class of diagram $D$ together
with a total order of $\partial_{a}D:=\partial D\cap a$ for each boundary arc
$a$, define uniquely an isotopy class of tangle. When choosing an orientation
$\mathfrak{o}(a)$ of a boundary arc $a$ and a diagram $D$, the set
$\partial_{a}D$ receives a natural order by setting that the points are
increasing when going in the direction of $\mathfrak{o}(a)$. We will represent
tangles by drawing a diagram and an orientation (an arrow) for each boundary
arc, as in Figure 1. When a boundary arc $a$ is oriented we assume that
$\partial_{a}D$ is ordered according to the orientation. A state of a tangle
is a map $s:\partial T\rightarrow\\{-,+\\}$. A pair $(T,s)$ is called a stated
tangle. We define a stated diagram $(D,s)$ in a similar manner.

Figure 1. On the left: a stated tangle. On the right: its associated diagram.
The arrows represent the height orders.

Let $A^{1/2}\in k^{\times}$ an invertible element and denote by $A$ its
square. Stated skein algebras were first introduced by Bonahon and Wong in
[BW11]. The version we will present here is a refinement due to Lê [Le18].

######  Definition 2.4.

[Le18] The stated skein algebra $\mathcal{S}_{A}(\mathbf{\Sigma})$ is the free
$k$-module generated by isotopy classes of stated tangles in
$\Sigma\times(0,1)$ modulo the following relations (1) and (2),

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The product of two classes of stated tangles $[T_{1},s_{1}]$ and
$[T_{2},s_{2}]$ is defined by isotoping $T_{1}$ and $T_{2}$ in
$\Sigma\times(1/2,1)$ and $\Sigma\times(0,1/2)$ respectively and then setting
$[T_{1},s_{1}]\cdot[T_{2},s_{2}]=[T_{1}\cup T_{2},s_{1}\cup s_{2}]$. Figure 2
illustrates this product.

For an unmarked surface, $\mathcal{S}_{A}(\Sigma,\emptyset)$ coincides with
the usual Kauffman-bracket skein algebra.

Figure 2. An illustration of the product in stated skein algebras.

Functoriality

Now consider an embedding $f:\mathbf{\Sigma}_{1}\to\mathbf{\Sigma}_{2}$ of
marked surfaces and define a proper embedding
$\widetilde{f}:\Sigma_{1}\times(0,1)\to\Sigma_{2}\times(0,1)$ such that: $(1)$
$\widetilde{f}(x,t)=(f(x),\varphi(x,t))$ for $\varphi$ a smooth map and $(2)$
if $a_{1},a_{2}$ are two boundary arcs of $\mathbf{\Sigma}_{1}$ mapped to the
same boundary arc $b$ of $\mathbf{\Sigma}_{2}$ and the ordering of $f$ is
$a_{1}<a_{2}$, then for all $x_{1}\in a_{1},x_{2}\in
a_{2},t_{1},t_{2}\in(0,1)$ one has
$\varphi(x_{1},t_{1})<\varphi(x_{2},t_{2})$. It is an easy consequence of the
definition that the formula
$f_{*}([T,s]):=(\widetilde{f}(T),s\circ\widetilde{f}^{-1})$ defines a morphism
of algebras
$f_{*}:\mathcal{S}_{A}(\mathbf{\Sigma}_{1})\to\mathcal{S}_{A}(\mathbf{\Sigma}_{2})$
independent on the choice of $\widetilde{f}$ and that the assignment
$\mathbf{\Sigma}\to\mathcal{S}_{A}(\mathbf{\Sigma})$ defines a symmetric
monoidal functor

$\mathcal{S}_{A}:\mathrm{MS}\to\mathrm{Alg}_{k}.$

Here is an alternative easier $2$-dimensional definition of $f_{*}$. Suppose
we have fixed some orientations $\mathfrak{o_{1}},\mathfrak{o_{2}}$ of the
boundary arcs of $\mathbf{\Sigma}_{1}$ and $\mathbf{\Sigma}_{2}$ and that
$f:\Sigma_{1}\to\Sigma_{2}$ is a proper oriented embedding sending
$\mathcal{A}_{1}$ to $\mathcal{A}_{2}$ in such a way that it preserves the
orientations of the boundary arcs. If $a_{1},a_{2}\in\mathcal{A}_{1}$ are sent
to the same boundary arc $b\in\mathcal{A}_{2}$, they are naturally ordered by
$<_{b}$, so $f$ together with $\mathfrak{o_{1}},\mathfrak{o}_{2}$ defines a
morphism in $\mathrm{MS}$ and we can choose $\widetilde{f}:=f\times id$, so
$f_{*}$ is defined on stated diagrams by $f_{*}([D,s]):=[f(D),s\circ f^{-1}]$.

Bases

The first theorem we state is the existence of bases for stated skein
algebras. Whereas proving the existence of bases for usual Kauffman-bracket
skein algebras (given by multicurves) is an easy exercise, the construction of
bases for marked surfaces is a highly non trivial result based on the Diamond
lemma.

A closed component of a diagram $D$ is trivial if it bounds an embedded disc
in $\Sigma$. An open component of $D$ is trivial if it can be isotoped,
relatively to its boundary, inside some boundary arc. A diagram is simple if
it has neither double point nor trivial component. By convention, the empty
set is a simple diagram. Let $\mathfrak{o}$ denote an arbitrary orientation of
the boundary arcs of $\mathbf{\Sigma}$. For each boundary arc $a$ we write
$<_{\mathfrak{o}}$ the induced total order on $\partial_{a}D$. A state
$s:\partial D\rightarrow\\{-,+\\}$ is $\mathfrak{o}-$increasing if for any
boundary arc $a$ and any two points $x,y\in\partial_{a}D$, then
$x<_{\mathfrak{o}}y$ implies $s(x)<s(y)$, with the convention $-<+$.

######  Definition 2.5.

We denote by
$\mathcal{B}^{\mathfrak{o}}\subset\mathcal{S}_{A}(\mathbf{\Sigma})$ the set of
classes of stated diagrams $(D,s)$ such that $D$ is simple and $s$ is
$\mathfrak{o}$-increasing.

######  Theorem 2.6.

(Lê [Le18, Theorem $2.11$]) The set $\mathcal{B}^{\mathfrak{o}}$ is a basis of
$\mathcal{S}_{A}(\mathbf{\Sigma})$.

The basis $\mathcal{B}^{\mathfrak{o}}$ is independent on the choice of the
ring $k$ and of $A^{1/2}\in k^{\times}$, in particular when
$k=\mathbb{Z}[A^{\pm 1/2}]$, the corresponding stated skein algebra is flat.
This fact will have deep consequences when defining deformation quantizations
of relative character varieties.

Off puncture ideal An easy but useful consequence of Theorem 2.6 is the
following. Suppose that $\mathbf{\Sigma}^{\prime}$ is obtained from
$\mathbf{\Sigma}$ by removing a puncture $p$, that is either by filling a
closed unmarked component of $\partial\Sigma$ (inner puncture) or fusioning
two adjacent boundary arcs (boundary puncture) to a single one (in which case
we need to order the two boundary arcs). Then the inclusion
$\Sigma\subset\Sigma^{\prime}$ defines a morphism
$f:\mathbf{\Sigma}\to\mathbf{\Sigma}^{\prime}$. Since any diagram in
$\Sigma^{\prime}$ is isotopic to a diagram in $\Sigma^{\prime}$, the morphism
$f_{*}:\mathcal{S}_{A}(\mathbf{\Sigma})\to\mathcal{S}_{A}(\mathbf{\Sigma}^{\prime})$
is surjective. The off-puncture ideal is
$\mathcal{I}_{p}:=\mathrm{ker}(f_{*})$, so we have an exact sequence:

(3)
$0\to\mathcal{I}_{p}\to\mathcal{S}_{A}(\mathbf{\Sigma})\xrightarrow{f_{*}}\mathcal{S}_{A}(\mathbf{\Sigma}^{\prime})\to
0.$

######  Proposition 2.7.

(K.-Quesney [KQ19a, Proposition $2.18$] ) The off puncture ideal
$\mathcal{I}_{p}$ is generated by the elements $[D_{1},s_{1}]-[D_{2},s_{2}]$,
where $(D_{1},s_{1})$ and $(D_{2},s_{2})$ are two connected stated diagrams
(so closed curves or stated arcs) in $\Sigma$ which are isotopic in
$\Sigma^{\prime}$.

Some useful skein relations Let us state some useful skein relations, which
are direct consequences of the definition. We first define some matrices:

$C=\begin{pmatrix}C_{+}^{+}&C_{-}^{+}\\\
C_{+}^{-}&C_{-}^{-}\end{pmatrix}:=\begin{pmatrix}0&A^{-1/2}\\\
-A^{-5/2}&0\end{pmatrix}.\mbox{ Therefore
}C^{-1}=-A^{3}C=\begin{pmatrix}0&-A^{5/2}\\\ A^{1/2}&0\end{pmatrix}.$

$\mathscr{R}=\begin{pmatrix}\mathscr{R}_{++}^{++}&\mathscr{R}_{+-}^{++}&\mathscr{R}_{-+}^{++}&\mathscr{R}_{--}^{++}\\\
\mathscr{R}_{++}^{+-}&\mathscr{R}_{+-}^{+-}&\mathscr{R}_{-+}^{+-}&\mathscr{R}_{--}^{+-}\\\
\mathscr{R}_{++}^{-+}&\mathscr{R}_{+-}^{-+}&\mathscr{R}_{-+}^{-+}&\mathscr{R}_{--}^{-+}\\\
\mathscr{R}_{++}^{--}&\mathscr{R}_{+-}^{--}&\mathscr{R}_{-+}^{--}&\mathscr{R}_{--}^{--}\end{pmatrix}:=\begin{pmatrix}A&0&0&0\\\
0&0&A^{-1}&0\\\ 0&A^{-1}&A-A^{-3}&0\\\ 0&0&0&A\end{pmatrix},\mbox{so
}\mathscr{R}^{-1}=\begin{pmatrix}A^{-1}&0&0&0\\\ 0&A^{-1}-A^{3}&A&0\\\
0&A&0&0\\\ 0&0&0&A^{-1}\end{pmatrix}.$

$\bullet$ The trivial arc relations:

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$\bullet$ The cutting arc relations:

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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@fill{0.7}\pgfsys@invoke{
}\color[rgb]{0.775,0.775,0.775}\definecolor[named]{pgfstrokecolor}{rgb}{0.775,0.775,0.775}\pgfsys@color@gray@stroke{0.775}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.775}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.775,0.775,0.775}{}\pgfsys@moveto{-9.95842pt}{-10.66978pt}\pgfsys@moveto{-9.95842pt}{-10.66978pt}\pgfsys@lineto{-9.95842pt}{10.66978pt}\pgfsys@lineto{5.69046pt}{10.66978pt}\pgfsys@lineto{5.69046pt}{-10.66978pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{10.66978pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{-9.95842pt}{-10.66978pt}\pgfsys@lineto{-9.95842pt}{8.66982pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{-9.95842pt}{8.66982pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{-4.26794pt}\pgfsys@lineto{-9.95842pt}{-4.26794pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{4.26794pt}\pgfsys@lineto{-9.95842pt}{4.26794pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.43217pt}{1.95961pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize{$i$}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{-14.22638pt}{-4.26794pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize{$j$}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\hskip
11.38092pt\leavevmode\hbox to16.45pt{\vbox
to21.74pt{\pgfpicture\makeatletter\hbox{\hskip
10.55841pt\lower-10.86978pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@gray@fill{0.775}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}{}\pgfsys@moveto{5.69046pt}{-10.66978pt}\pgfsys@lineto{5.69046pt}{10.66978pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-9.95842pt}{-4.26794pt}\pgfsys@lineto{-5.69046pt}{-4.26794pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-9.95842pt}{4.26794pt}\pgfsys@lineto{-5.69046pt}{4.26794pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}{}}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}} {{}{}{}{}}}}{}
{} {} {} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.15pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-5.69046pt}{0.0pt}\pgfsys@moveto{-5.69046pt}{-4.26794pt}\pgfsys@curveto{-3.33331pt}{-4.26794pt}{-1.42252pt}{-2.35715pt}{-1.42252pt}{0.0pt}\pgfsys@curveto{-1.42252pt}{2.35715pt}{-3.33331pt}{4.26794pt}{-5.69046pt}{4.26794pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\sum_{i,j=\pm}(C^{-1})_{j}^{i}\hskip
5.69046pt\leavevmode\hbox to24.78pt{\vbox
to21.74pt{\pgfpicture\makeatletter\hbox{\hskip
10.55841pt\lower-10.86978pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{5.69046pt}{-10.66978pt}\pgfsys@lineto{5.69046pt}{8.66982pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{-4.26794pt}\pgfsys@lineto{-9.95842pt}{-4.26794pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{5.69046pt}{4.26794pt}\pgfsys@lineto{-9.95842pt}{4.26794pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}\pgfsys@moveto{9.24706pt}{4.26794pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.04126pt}{1.95961pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize{$i$}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{9.24706pt}{-4.26794pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.6054pt}{-5.89574pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

.

$\bullet$ The height exchange relations:

(6) $\leavevmode\hbox to24.78pt{\vbox
to21.74pt{\pgfpicture\makeatletter\hbox{\hskip
10.55841pt\lower-10.86978pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@fill{0.7}\pgfsys@invoke{
}\color[rgb]{0.775,0.775,0.775}\definecolor[named]{pgfstrokecolor}{rgb}{0.775,0.775,0.775}\pgfsys@color@gray@stroke{0.775}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.775}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.775,0.775,0.775}{}\pgfsys@moveto{-9.95842pt}{-10.66978pt}\pgfsys@moveto{-9.95842pt}{-10.66978pt}\pgfsys@lineto{-9.95842pt}{10.66978pt}\pgfsys@lineto{5.69046pt}{10.66978pt}\pgfsys@lineto{5.69046pt}{-10.66978pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{10.66978pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{5.69046pt}{-10.66978pt}\pgfsys@lineto{5.69046pt}{8.66982pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{5.69046pt}{8.66982pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{-4.26794pt}\pgfsys@lineto{-9.95842pt}{-4.26794pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{4.26794pt}\pgfsys@lineto{-9.95842pt}{4.26794pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}\pgfsys@moveto{9.24706pt}{4.26794pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize{$i$}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{9.24706pt}{-4.26794pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.6054pt}{-5.89574pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize{$j$}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=25.6073pt]{cross2.eps}}}}}}}$}=\sum_{k,l=\pm}\mathscr{R}_{ij}^{kl}\hskip
5.69046pt\leavevmode\hbox to25.07pt{\vbox
to21.74pt{\pgfpicture\makeatletter\hbox{\hskip
10.55841pt\lower-10.86978pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.7,0.7,0.7}\pgfsys@color@gray@fill{0.7}\pgfsys@invoke{
}\color[rgb]{0.775,0.775,0.775}\definecolor[named]{pgfstrokecolor}{rgb}{0.775,0.775,0.775}\pgfsys@color@gray@stroke{0.775}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0.775}\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.775,0.775,0.775}{}\pgfsys@moveto{-9.95842pt}{-10.66978pt}\pgfsys@moveto{-9.95842pt}{-10.66978pt}\pgfsys@lineto{-9.95842pt}{10.66978pt}\pgfsys@lineto{5.69046pt}{10.66978pt}\pgfsys@lineto{5.69046pt}{-10.66978pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{10.66978pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}{}{}{}{}{{}}\pgfsys@moveto{5.69046pt}{-8.66982pt}\pgfsys@lineto{5.69046pt}{10.66978pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{5.69046pt}{-8.66982pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{-4.26794pt}\pgfsys@lineto{-9.95842pt}{-4.26794pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{4.26794pt}\pgfsys@lineto{-9.95842pt}{4.26794pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}\pgfsys@moveto{9.24706pt}{4.26794pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.13387pt}{1.8374pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize{$l$}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{9.24706pt}{-4.26794pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.31477pt}{-6.69849pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize{$k$}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\hskip
11.38092pt\leavevmode\hbox to24.78pt{\vbox
to21.74pt{\pgfpicture\makeatletter\hbox{\hskip
10.55841pt\lower-10.86978pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@gray@fill{0.775}\pgfsys@invoke{
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Graduations Fix an orientation $\mathfrak{o}$ of the boundary arcs of
$\mathbf{\Sigma}$. A simple diagram $D$ induces a relative homology class
$[D]\in\mathrm{H}_{1}(\Sigma,\mathcal{A};\mathbb{Z}/2\mathbb{Z})$.

######  Definition 2.8.

For $\chi\in\mathrm{H}_{1}(\Sigma,\mathcal{A};\mathbb{Z}/2\mathbb{Z})$, we
define the subspace:

$\mathcal{S}_{A}^{(\chi)}(\mathbf{\Sigma}):=\mathrm{Span}\left([D,s]\in\mathcal{B}^{\mathfrak{o}}|[D]=\chi\right)\subset\mathcal{S}_{A}(\mathbf{\Sigma}).$

It follows from the defining skein relations (1) and (2) that
$\mathcal{S}_{A}^{(\chi)}(\mathbf{\Sigma})$ does not depend on the choice of
$\mathfrak{o}$ and that
$\mathcal{S}_{A}(\mathbf{\Sigma})=\oplus_{\chi\in\mathrm{H}_{1}(\Sigma,\mathcal{A};\mathbb{Z}/2\mathbb{Z})}\mathcal{S}_{A}^{(\chi)}(\mathbf{\Sigma})$
is a graded algebra, i.e. that
$\mathcal{S}_{A}^{(\chi_{1})}(\mathbf{\Sigma})\cdot\mathcal{S}_{A}^{(\chi_{2})}(\mathbf{\Sigma})\subset\mathcal{S}_{A}^{(\chi_{1}+\chi_{2})}(\mathbf{\Sigma})$.

Reduced stated skein algebras

Let $\mathbf{\Sigma}$ a marked surface and $p$ a boundary puncture between two
consecutive boundary arcs $a$ and $b$ on the same boundary component
$\partial$ of $\partial\Sigma$. The orientation of $\Sigma$ induces an
orientation of $\partial$ so a cyclic ordering of the elements of
$\partial\cap\mathcal{A}$ we suppose that $b$ is followed by $a$ in this
ordering. We denote by $\alpha(p)$ an arc with one endpoint $v_{a}\in a$ and
one endpoint $v_{b}\in b$ such that $\alpha(p)$ can be isotoped inside
$\partial$. Let $\alpha(p)_{ij}\in\mathcal{S}_{A}(\mathbf{\Sigma})$ be the
class of the stated arc $(\alpha(p),s)$ where $s(v_{a})=i$ and $s(v_{b})=j$.
The following definition will be justified in the next section when studying
the quantum trace.

######  Definition 2.9.

We call bad arc associated to $p$ the element
$\alpha(p)_{-+}\in\mathcal{S}_{A}(\mathbf{\Sigma})$ (see Figure 3). The
reduced stated skein algebra $\mathcal{S}^{red}_{A}(\mathbf{\Sigma})$ is the
quotient of $\mathcal{S}_{A}(\mathbf{\Sigma})$ by the ideal generated by all
bad arcs.

Figure 3. A bad arc.

The orientation of $\Sigma$ induces an orientation $\mathfrak{o}^{+}$ of the
boundary arcs of $\mathcal{A}$. Let
$\overline{\mathcal{B}}^{\mathfrak{o}^{+}}\subset\mathcal{S}_{A}^{red}(\mathbf{\Sigma})$
be the set of classes $[D,s]^{\mathfrak{o}^{+}}$ where $D$ is simple, $s$ is
$\mathfrak{o}^{+}$-increasing and $(D,s)$ does not contain any bad arc.

######  Theorem 2.10.

(Costantino-Le [CL19, Theorem $7.1$]) The set
$\overline{\mathcal{B}}^{\mathfrak{o}^{+}}$ is a basis of
$\mathcal{S}^{red}_{A}(\mathbf{\Sigma})$.

###  2.3. The gluing/cutting/excision formula and triangular decomposition

The main interest in extending skein algebras to marked surfaces is their
behaviour for the gluing operation that we now describe.

Let $a$, $b$ be two distinct boundary arcs of $\mathbf{\Sigma}$, denote by
$\pi:\Sigma\rightarrow\Sigma_{a\\#b}$ the projection and $c:=\pi(a)=\pi(b)$.
Let $(T_{0},s_{0})$ be a stated framed tangle of $\Sigma_{a\\#b}\times(0,1)$
transversed to $c\times(0,1)$ and such that the heights of the points of
$T_{0}\cap c\times(0,1)$ are pairwise distinct and the framing of the points
of $T_{0}\cap c\times(0,1)$ is vertical towards $1$. Let
$T\subset\Sigma\times(0,1)$ be the framed tangle obtained by cutting $T_{0}$
along $c$. Any two states $s_{a}:\partial_{a}T\rightarrow\\{-,+\\}$ and
$s_{b}:\partial_{b}T\rightarrow\\{-,+\\}$ give rise to a state
$(s_{a},s,s_{b})$ on $T$. Both the sets $\partial_{a}T$ and $\partial_{b}T$
are in canonical bijection with the set $T_{0}\cap c$ by the map $\pi$. Hence
the two sets of states $s_{a}$ and $s_{b}$ are both in canonical bijection
with the set $\mathrm{St}(c):=\\{s:c\cap T_{0}\rightarrow\\{-,+\\}\\}$.

######  Definition 2.11.

Let
$\theta_{a\\#b}:\mathcal{S}_{A}(\mathbf{\Sigma}_{a\\#b})\rightarrow\mathcal{S}_{A}(\mathbf{\Sigma})$
be the linear map given, for any $(T_{0},s_{0})$ as above, by:

$\theta_{a\\#b}\left([T_{0},s_{0}]\right):=\sum_{s\in\mathrm{St}(c)}[T,(s,s_{0},s)].$

Figure 4. An illustration of the gluing map $\theta_{a\\#b}$.

######  Theorem 2.12.

( Lê [Le18, Theorem $3.1$]) The linear map
$\theta_{a\\#b}:\mathcal{S}_{A}(\mathbf{\Sigma}_{a\\#b})\rightarrow\mathcal{S}_{A}(\mathbf{\Sigma})$
is an injective morphism of algebras. Moreover the gluing operation is
coassociative in the sense that if $a,b,c,d$ are four distinct boundary arcs,
then we have
$\theta_{a\\#b}\circ\theta_{c\\#d}=\theta_{c\\#d}\circ\theta_{a\\#b}$.

Consider the bigon $\mathbb{B}$, that is a disc with two boundary arcs, say
$a_{L}$ and $a_{R}$. Take two copies $\mathbb{B}\bigsqcup\mathbb{B}^{\prime}$:
when gluing $a_{R}$ to $a_{L}^{\prime}$ we get another bigon, so we have a
map:

$\Delta:=\theta_{a_{R}\\#a_{L}^{\prime}}:\mathcal{S}_{A}(\mathbb{B})^{\otimes
2}\to\mathcal{S}_{A}(\mathbb{B}).$

The coassociativity of Theorem 2.12 implies that $\Delta$ is coassociative.
Denote by $\alpha_{ij}\in\mathcal{S}_{A}(\mathbb{B})$ the class of a single
arc $\alpha$ connecting $a_{L}$ to $a_{R}$ with state $i$ on $\alpha\cap
a_{L}$ and $j$ on $\alpha\cap a_{R}$. Lê proved in [Le18] that
$\mathcal{S}_{A}(\mathbb{B})$ is generated by the $\alpha_{ij}$ together with
the following relations, where we state $q:=A^{2}$:

(7) $\displaystyle\alpha_{++}\alpha_{+-}$
$\displaystyle=q^{-1}\alpha_{+-}\alpha_{++}$
$\displaystyle\alpha_{++}\alpha_{-+}$
$\displaystyle=q^{-1}\alpha_{-+}\alpha_{++}$ (8)
$\displaystyle\alpha_{--}\alpha_{+-}$ $\displaystyle=q\alpha_{+-}\alpha_{--}$
$\displaystyle\alpha_{--}\alpha_{-+}$ $\displaystyle=q\alpha_{-+}\alpha_{--}$
(9) $\displaystyle\alpha_{++}\alpha_{--}$
$\displaystyle=1+q^{-1}\alpha_{+-}\alpha_{-+}$
$\displaystyle\alpha_{--}\alpha_{++}$
$\displaystyle=1+q\alpha_{+-}\alpha_{-+}$ (10)
$\displaystyle\alpha_{-+}\alpha_{+-}$ $\displaystyle=\alpha_{+-}\alpha_{-+}$

We turn $\mathcal{S}_{A}(\mathbb{B})$ into a Hopf algebra using the counit and
antipode given by

$\begin{pmatrix}\epsilon(\alpha_{++})&\epsilon(\alpha_{+-})\\\
\epsilon(\alpha_{-+})&\epsilon(\alpha_{--})\end{pmatrix}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\text{ and }\begin{pmatrix}S(\alpha_{++})&S(\alpha_{+-})\\\
S(\alpha_{-+})&S(\alpha_{--})\end{pmatrix}=\begin{pmatrix}\alpha_{--}&-q\alpha_{+-}\\\
-q^{-1}\alpha_{-+}&\alpha_{++}\end{pmatrix}.$

######  Notations 2.13.

We denote by $\mathcal{O}_{q}[\operatorname{SL}_{2}]$ the Hopf algebra
$(\mathcal{S}_{A}(\mathbb{B}),\Delta,\epsilon,S)$.

Now consider a marked surface $\mathbf{\Sigma}$ and a boundary arc $c$. By
gluing a bigon $\mathbb{B}$ along $\mathbf{\Sigma}$ while gluing $b$ with $c$,
one obtains a punctured surface isomorphic to $\mathbf{\Sigma}$, hence a map

$\Delta_{c}^{L}:=\theta_{b\\#c}:\mathcal{S}_{A}(\mathbf{\Sigma})\rightarrow\mathcal{O}_{q}[\operatorname{SL}_{2}]\otimes\mathcal{S}_{A}(\mathbf{\Sigma})$

which endows $\mathcal{S}_{A}(\mathbf{\Sigma})$ with a structure of left
$\mathcal{O}_{q}[\operatorname{SL}_{2}]$ comodule (see Figure 5). Similarly,
gluing $c$ with $a$ induces a right comodule morphism
$\Delta_{c}^{R}:=\theta_{c\\#a}:\mathcal{S}_{A}(\mathbf{\Sigma})\rightarrow\mathcal{S}_{A}(\mathbf{\Sigma})\otimes\mathcal{O}_{q}[\operatorname{SL}_{2}]$.

Figure 5. An illustration of the coproduct $\Delta$ in
$\mathcal{O}_{q}[\operatorname{SL}_{2}]$ and of the left comodule map
$\Delta_{a}^{L}$.

Probably the most fundamental property which justifies the importance and
usefulness of stated skein algebra is the following:

######  Theorem 2.14.

(K.-Quesney [KQ19a, Theorem $1.1$], Costantino-Lê [CL19, Theorem $4.7$], see
also Cooke [Coo19] in a different language) Let $\mathbf{\Sigma}$ be a marked
surface and $a,b$ two boundary arcs. The following sequence is exact:

$0\to\mathcal{S}_{A}(\mathbf{\Sigma}_{a\\#b})\xrightarrow{\theta_{a\\#b}}\mathcal{S}_{A}(\mathbf{\Sigma})\xrightarrow{\Delta^{L}_{a}-\sigma\circ\Delta^{R}_{b}}\mathcal{O}_{q}[\operatorname{SL}_{2}]\otimes\mathcal{S}_{A}(\mathbf{\Sigma}),$

where $\sigma(x\otimes y):=y\otimes x$.

In particular, when $\mathbf{\Sigma}$ is equipped with a triangulation
$\mathbb{T}$, i.e. when $\mathbf{\Sigma}$ is obtained from a disjoint union of
triangles by gluing some pairs $(e_{i},e_{i}^{\prime})$ of edges, by composing
the maps $\theta_{e_{i}\\#e_{i}^{\prime}}$, we get an injective map:

$\theta^{\Delta}:\mathcal{S}_{A}(\mathbf{\Sigma})\hookrightarrow\otimes_{\mathbb{T}\in
F(\Delta)}\mathcal{S}_{A}(\mathbb{T}).$

Also, by composing the comodule maps $\Delta^{L}_{e_{i}\\#e_{i}^{\prime}}$,
one gets a Hopf comodule map

$\Delta^{L}:\otimes_{\mathbb{T}\in
F(\Delta)}\mathcal{S}_{A}(\mathbb{T})\rightarrow\left(\otimes_{e\in\mathring{\mathcal{E}}(\Delta)}\mathcal{O}_{q}[\operatorname{SL}_{2}]\right)\otimes\left(\otimes_{\mathbb{T}\in
F(\Delta)}\mathcal{S}_{A}(\mathbb{T})\right).$

and a right comodule map

$\Delta^{R}:\otimes_{\mathbb{T}\in
F(\Delta)}\mathcal{S}_{A}(\mathbb{T})\rightarrow\left(\otimes_{\mathbb{T}\in
F(\Delta)}\mathcal{S}_{A}(\mathbb{T})\right)\otimes\left(\otimes_{e\in\mathring{\mathcal{E}}(\Delta)}\mathcal{O}_{q}[\operatorname{SL}_{2}]\right).$

######  Corollary 2.15.

The following sequence is exact:

(11)
$0\rightarrow\mathcal{S}_{A}(\mathbf{\Sigma})\xrightarrow{\theta^{\Delta}}\otimes_{\mathbb{T}\in
F(\Delta)}\mathcal{S}_{A}(\mathbb{T})\xrightarrow{\Delta^{L}-\sigma\circ\Delta^{R}}\left(\otimes_{e\in\mathring{\mathcal{E}}(\Delta)}\mathcal{O}_{q}[\operatorname{SL}_{2}]\right)\otimes\left(\otimes_{\mathbb{T}\in
F(\Delta)}\mathcal{S}_{A}(\mathbb{T})\right).$

For further use, let us emphasize a second consequence of the gluing property.
Let $\mathbf{\Sigma}=(\Sigma,\mathcal{A})$ and let
$\mathcal{B}\subset\mathcal{A}$ a subset of boundary arcs and consider
$\underline{\mathbf{\Sigma}}:=(\Sigma,\mathcal{A}\setminus\mathcal{B})$
obtained by removing the boundary arcs of $\mathcal{B}$. The identity map of
$\Sigma$ defines an embedding $\underline{\mathbf{\Sigma}}\to\mathbf{\Sigma}$,
so a morphism
$\iota:\mathcal{S}_{A}(\underline{\mathbf{\Sigma}})\to\mathcal{S}_{A}(\mathbf{\Sigma})$.
Note that $\underline{\mathbf{\Sigma}}$ is obtained from $\mathbf{\Sigma}$ by
gluing a monogon $\mathbf{m}_{0}$ to each boundary arc of $\mathcal{B}$. After
identifying $\mathcal{S}_{A}(\mathbf{m}_{0})\cong k$ through the isomorphism
sending the empty link to $1\in k$, $\iota$ can be alternatively defined as
the gluing map:

$\iota:\mathcal{S}_{A}(\underline{\mathbf{\Sigma}})\xrightarrow{\theta}\mathcal{S}_{A}(\mathbf{\Sigma}\cup\mathbf{m}_{0}^{\cup\mathcal{B}})\cong\left(\mathcal{S}_{A}(\mathbf{m}_{0})\right)^{\otimes\mathbb{B}}\otimes\mathcal{S}_{A}(\mathbf{\Sigma})\cong\mathcal{S}_{A}(\mathbf{\Sigma}).$

From Theorem 2.14, we get an exact sequence:

$0\to\mathcal{S}_{A}(\underline{\mathbf{\Sigma}})\xrightarrow{\theta}\mathcal{S}_{A}(\mathbf{\Sigma}\cup\mathbf{m}_{0}^{\cup\mathcal{B}})\xrightarrow{\Delta^{L}-\sigma\circ\Delta^{R}}\left(\mathcal{O}_{q}[\operatorname{SL}_{2}]\right)^{\otimes\mathcal{B}}\otimes\mathcal{S}_{A}(\mathbf{\Sigma}\cup\mathbf{m}_{0}^{\cup\mathcal{B}}).$

Using the identification
$\mathcal{S}_{A}(\mathbf{\Sigma}\cup\mathbf{m}_{0}^{\cup\mathcal{B}})\cong\mathcal{S}_{A}(\mathbf{\Sigma})$,
the comodule maps $\Delta^{L},\Delta^{R}$ induces comodules maps
$\Delta^{L}_{\mathcal{B}}:\mathcal{S}_{A}(\mathbf{\Sigma})\to\left(\mathcal{O}_{q}[\operatorname{SL}_{2}]\right)^{\otimes\mathcal{B}}\otimes\mathcal{S}_{A}(\mathbf{\Sigma})$
and
$\Delta^{R}_{\mathcal{B}}:\mathcal{S}_{A}(\mathbf{\Sigma})\to\mathcal{S}_{A}(\mathbf{\Sigma})\otimes\left(\mathcal{O}_{q}[\operatorname{SL}_{2}]\right)^{\otimes\mathcal{B}}$.
By definition, $\Delta^{R}$ is defined by gluing some bigons $\mathbb{B}$ to
the monogons $\mathbf{m}_{0}$. While gluing a bigon to a monogon, we get
another monogon and using the identification
$\mathcal{S}_{A}(\mathbf{m}_{0})\cong k$, the gluing map
$\mathcal{S}_{A}(\mathbf{m}_{0})\to\mathcal{S}_{A}(\mathbf{m_{0}}\cup\mathbb{B})$
corresponds to the unit map $\eta:k\to\mathcal{O}_{q}[\operatorname{SL}_{2}]$,
therefore
$\Delta^{R}_{\mathcal{B}}=id\otimes(\eta\circ\epsilon)^{\mathcal{B}}$ and we
get the

######  Corollary 2.16.

The following sequence is exact:

$0\to\mathcal{S}_{A}(\underline{\mathbf{\Sigma}})\xrightarrow{\iota}\mathcal{S}_{A}(\mathbf{\Sigma})\xrightarrow{\Delta^{L}_{\mathcal{B}}-(\eta\circ\epsilon)^{\mathcal{B}}\otimes
id}\left(\mathcal{O}_{q}[\operatorname{SL}_{2}]\right)^{\otimes\mathcal{B}}\otimes\mathcal{S}_{A}(\mathbf{\Sigma}).$

Said differently,
$\mathcal{S}_{A}(\underline{\mathbf{\Sigma}})\subset\mathcal{S}_{A}(\mathbf{\Sigma})$
is the subalgebra of coinvariant vectors for the left co-action
$\Delta^{L}_{\mathcal{B}}$.

###  2.4. The quantum fusion operation

The fusion operation was introduced by Alekseev and Malkin in the study of
relative character varieties. Its interpretation in terms of marked surfaces
and skein algebras was made by Costantino and Lê. Recall that the triangle
$\mathbb{T}$ is a disc with three boundary arcs say $e_{1},e_{2},e_{3}$.

######  Definition 2.17.

Let $\mathbf{\Sigma}$ a marked surface with some boundary arcs $a$ and $b$.

The fusion of  $a$ and $b$ is the marked surface
$\mathbf{\Sigma}_{a\circledast b}$ obtained from
$\mathbf{\Sigma}\bigsqcup\mathbb{T}$ by gluing $(a,e_{1})$ and $(b,e_{2})$,
i.e.

$\mathbf{\Sigma}_{a\circledast
b}:=\left(\mathbf{\Sigma}\bigsqcup\mathbb{T}\right)_{a\\#e_{1},b\\#e_{2}}.$

######  Example 2.18.

  1. (1)

the triangle $\mathbb{T}$ is obtained from
$\mathbb{B}\bigsqcup\mathbb{B}^{\prime}$ by fusioning $a_{R}$ and
$a_{L}^{\prime}$. We write $\mathbb{T}=\mathbb{B}\circledast\mathbb{B}$.

  2. (2)

$\mathbf{m}_{1}$ is obtained from $\mathbb{B}$ by fusioning its two boundary
arcs.

  3. (3)

While fusioning the two boundary arcs of
$\mathbf{\Sigma}_{g_{1},n_{1}}^{0}\bigsqcup\mathbf{\Sigma}_{g_{2},n_{2}}^{0}$
we get $\mathbf{\Sigma}^{0}_{g_{1}+g_{2},n_{1}+n_{2}}$. We thus write:

$\mathbf{\Sigma}_{g,n}^{0}={\mathbf{\Sigma}^{0}_{1,0}}^{\circledast
g}\circledast\mathbf{m}_{1}^{\circledast n}.$

In order to understand the fusion operation at the level of stated skein
algebras, we need to endow $\mathcal{O}_{q}[\operatorname{SL}_{2}]$ with a
structure of cobraided Hopf algebra. Define a co-R matrix
$\mathbf{r}:\mathcal{O}_{q}[\operatorname{SL}_{2}]^{\otimes 2}\to k$ by the
formula

$\mathbf{r}\left((\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle1.eps}}}}}}}$}\otimes(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle2.eps}}}}}}}$}\right):=\epsilon\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{TangleBraiding.eps}}}}}}}$}\right),$

where a letter $n$ on top of a strand means that we take $n$ parallel copies.
Note that the height exchange formula (6) implies that $\mathbf{r}$ is
characterized by

$\mathbf{r}(\alpha_{ij}\otimes\alpha_{kl})=\mathscr{R}^{jl}_{ki}.$

Recall that, by definition, a bilinear form
$\mathbf{r}:\mathcal{O}_{q}[\operatorname{SL}_{2}]^{\otimes 2}\to k$ is a co-R
matrix means that there exists another linear form
$\overline{\mathbf{r}}:\mathcal{O}_{q}[\operatorname{SL}_{2}]^{\otimes 2}\to
k$ it satisfies the axioms

  1. ($E_{1}$)

$\mathbf{r}\star\overline{\mathbf{r}}=\overline{\mathbf{r}}\star
r=\epsilon\otimes\epsilon$;

  2. ($E_{2}$)

$\mu^{op}=\mathbf{r}\star\mu\star\overline{\mathbf{r}}$;

  3. ($E_{3}$)

$\mathbf{r}(\mu\otimes id)=\mathbf{r}_{13}\star\mathbf{r}_{23}$ and
$\mathbf{r}(id\otimes\mu)=\mathbf{r}_{13}\star\mathbf{r}_{12}$.

where $\star$ is the convolution operation on linear forms and
$\mathbf{r}_{12}:=\mathbf{r}\otimes\epsilon,\mathbf{r}_{23}:=\epsilon\otimes\mathbf{r},\mathbf{r}_{13}:=(\epsilon\otimes\mathbf{r})(\tau\otimes
id)$. Define $\overline{\mathbf{r}}$ by:

$\overline{\mathbf{r}}\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle1.eps}}}}}}}$}\otimes(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle2.eps}}}}}}}$}\right):=\epsilon\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{TangleBraidingBar.eps}}}}}}}$}\right).$

Said differently,
$\overline{\mathbf{r}}(\alpha_{ij}\otimes\alpha{kl})=(\mathscr{R}^{-1})^{jl}_{ki}$.
The axioms are proved graphically as follows:

Axiom $(E_{1})$:

$(\mathbf{r}\star\overline{\mathbf{r}})\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle1.eps}}}}}}}$}\otimes\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle2.eps}}}}}}}$}\right)=\sum_{i,j}\epsilon\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{R11.eps}}}}}}}$}\right)\epsilon\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{R12.eps}}}}}}}$}\right)=\epsilon\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=42.67912pt]{R13.eps}}}}}}}$}\right)\\\
=\epsilon\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{R14.eps}}}}}}}$}\right)=(\epsilon\otimes\epsilon)\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle1.eps}}}}}}}$}\otimes\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle2.eps}}}}}}}$}\right).$

The equality $\overline{\mathbf{r}}\star\mathbf{r}=\epsilon\otimes\epsilon$ is
proved similarly.
Axiom $(E_{2})$:

$(\mathbf{r}\star\mu\star\overline{\mathbf{r}})\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle1.eps}}}}}}}$}\otimes\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle2.eps}}}}}}}$}\right)=\sum_{i,j,k,l}\epsilon\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{R11.eps}}}}}}}$}\right)\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{R21.eps}}}}}}}$}\epsilon\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{R22.eps}}}}}}}$}\right)\\\
=\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{R23.eps}}}}}}}$}=\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{R24.eps}}}}}}}$}=\mu^{op}\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle1.eps}}}}}}}$}\otimes\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle2.eps}}}}}}}$}\right).$

Axiom $(E_{3})$:

$\mathbf{r}(\mu\otimes
id)\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle0.eps}}}}}}}$}\otimes\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle1.eps}}}}}}}$}\otimes\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle2.eps}}}}}}}$}\right)=\epsilon\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{R31.eps}}}}}}}$}\right)\\\
\sum_{i,j,k}\epsilon\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{R32.eps}}}}}}}$}\right)\epsilon\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{R33.eps}}}}}}}$}\right)=r_{13}\star
r_{23}\left(\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle0.eps}}}}}}}$}\otimes\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle1.eps}}}}}}}$}\otimes\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{\includegraphics[width=36.98866pt]{Tangle2.eps}}}}}}}$}\right).$

The equality
$\mathbf{r}\star(id\otimes\mu)=\mathbf{r}_{13}\star\mathbf{r}_{12}$ is proved
similarly.

When $(H,\mu,\eta,\Delta,\epsilon,S,\mathbf{r})$ is a cobraided Hopf algebra,
its category $H-\mathrm{Comod}$ of finite dimensional left comodules has a
natural structure of braided category where for $(V,\Delta_{V})$ and
$(W,\Delta_{W})$ two $H$-comodules, the braiding is given by:

$c_{V,W}:V\otimes W\xrightarrow{\tau_{V,W}}W\otimes
V\xrightarrow{\Delta_{W}\otimes\Delta_{V}}H\otimes W\otimes H\otimes
V\xrightarrow{id_{H}\otimes\tau_{W,H}\otimes id_{V}}H\otimes H\otimes W\otimes
V\xrightarrow{r\otimes id_{W}\otimes id_{V}}W\otimes V.$

######  Definition 2.19.

Let $(H,\mu_{H},\eta,\Delta,\epsilon_{H},S,\mathbf{r})$ be a cobraided Hopf
algebra and $\mathbf{A}=(A,\mu,\epsilon)$ be an algebra object in $H\otimes
H-\mathrm{Comod}$ and denote by $\Delta_{H\otimes H}:A\to H\otimes H\otimes A$
its comodule map. Write $\Delta^{1}:=(id\otimes\epsilon\otimes
id)\circ\Delta_{H\otimes H}:A\to H\otimes A$ and $\Delta^{2}:(\epsilon\otimes
id\otimes id)\circ\Delta_{H\otimes H}:A\to H\otimes A$. The fusion
$\mathbf{A}_{1\circledast 2}$ is the algebra object $(A_{1\circledast
2},\mu_{1\circledast 2},\epsilon_{1\circledast 2})$ in $H-\mathrm{Comod}$
where

  1. (1)

$A_{1\circledast 2}=A$ as a $k$-module and $\epsilon_{1\circledast
2}=\epsilon$;

  2. (2)

The product is the composition

$\mu_{1\circledast 2}:A\otimes
A\xrightarrow{\Delta_{1}\otimes\Delta_{2}}H\otimes A\otimes H\otimes
A\xrightarrow{id\otimes\tau_{A,H}\otimes id}H\otimes H\otimes A\otimes
A\xrightarrow{\mathbf{r}\circ\tau_{H,H}\otimes\mu}A.$

  3. (3)

The comodule map is $\Delta_{H}:=(\mu_{H}\otimes id)\circ\Delta_{H\otimes H}$.

For instance, if $V$ and $W$ are two algebra objects in $H-\mathrm{Comod}$,
then $V\otimes_{k}W$ is an algebra object in $H^{\otimes 2}-\mathrm{Comod}$
and its fusion $V\circledast W$ is called the cobraided tensor product.
Identify $V$ with $V\otimes 1$ and $W$ with $1\otimes W$ in $V\otimes W$. Its
product is characterized by the formula

$\mu(x\otimes y)=\left\\{\begin{array}[]{ll}\mu_{V}(x\otimes y)&\mbox{, if
}x,y\in V;\\\ \mu_{W}(x\otimes y)&\mbox{, if }x,y\in W;\\\ x\otimes y&\mbox{,
if }x\in V,y\in W;\\\ c_{W,V}(x\otimes y)&\mbox{, if }x\in W,y\in
V.\end{array}\right.$

Now consider a marked surface $\mathbf{\Sigma}_{a\circledast b}$ obtained by
fusioning two boundary arcs $a$ and $b$. Fix an orientation $\mathfrak{o}$ of
the boundary arcs of $\mathbf{\Sigma}\bigsqcup\mathbb{T}$, as in Figure 6 and
let $\mathfrak{o}^{\prime}$ the induced orientation of the boundary arcs of
$\mathbf{\Sigma}^{\prime}$. Define a linear map $\Psi_{a\circledast
b}:\mathcal{S}_{A}(\mathbf{\Sigma})\to\mathcal{S}_{A}(\mathbf{\Sigma}_{a\\#b})$
by $\Psi_{a\circledast b}([D,s]^{\mathfrak{o}}):=[D^{\prime},s^{\prime}]$
where $(D^{\prime},s^{\prime})$ is obtained from $(D,s)$ by gluing to each
point of $D\cap a$ a straight line in $\mathbb{T}$ between $e_{1}$ and $e_{3}$
and by gluing to each point of $D\cap b$ a straight line in $\mathbb{T}$
between $e_{2}$ and $e_{3}$. Figure 6 illustrates $\Psi_{a\circledast b}$.

Figure 6. An illustration of $\Psi_{a\circledast b}$.

######  Theorem 2.20.

(Costantino-Lê [CL19, Theorem $4.13$]) The linear map $\Psi_{a\circledast b}$
is an isomorphism of $k$-modules which identifies
$\mathcal{S}_{A}(\mathbf{\Sigma}_{a\circledast b})$ with the fusion
$\mathcal{S}_{A}(\mathbf{\Sigma})_{a\circledast b}$.

###### Proof (Higgins).

The fact that $\Psi_{a\circledast b}$ is surjective is an easy consequence of
the cutting arc relation (5). The injectivity is proved using the following
elegant argument of Higgins in [Hig]. Since $\mathbf{\Sigma}_{a\circledast b}$
is obtained from $\mathbf{\Sigma}\bigsqcup\mathbb{T}$ by gluing some boundary
arcs, we have a gluing map
$\theta:\mathcal{S}_{A}(\mathbf{\Sigma}_{a\circledast
b})\to\mathcal{S}_{A}(\mathbb{T})\otimes\mathcal{S}_{A}(\mathbf{\Sigma})$. Let
$i:\mathbb{T}\to\mathbb{B}$ be the embedding of marked surfaces sending
$e_{3}$ to $a_{R}$ and $e_{1},e_{2}$ to $a_{L}$ with $e_{1}>e_{2}$ and denote
by $i_{*}:\mathcal{S}_{A}\to\mathcal{O}_{q}[\operatorname{SL}_{2}]$ the
induced morphism. Consider the composition

$\Phi:\mathcal{S}_{A}(\mathbf{\Sigma}_{a\circledast
b})\xrightarrow{\theta}\mathcal{S}_{A}(\mathbb{T})\otimes\mathcal{S}_{A}(\mathbf{\Sigma})\xrightarrow{i_{*}\otimes
id}\mathcal{O}_{q}[\operatorname{SL}_{2}]\otimes\mathcal{S}_{A}(\mathbf{\Sigma})\xrightarrow{\epsilon\otimes
id}\mathcal{S}_{A}(\mathbf{\Sigma}).$

As illustrated in Figure 7, it is easy to see that $\Phi$ is a left inverse to
$\Psi_{a\circledast b}$, thus $\Psi_{a\circledast b}$ is an isomorphism. It
remains to prove that the pullback by $\Psi_{a\circledast b}$ of the product
in $\mathcal{S}_{A}(\mathbf{\Sigma}_{a\circledast b})$ is the fusion product
$\mu_{a\circledast b}$. This fact is illustrated in Figure 8.

Figure 7. An illustration of the equality $\Phi\circ\Psi_{a\circledast b}=id$.

Figure 8. An illustration of the fusion product $\mu_{\circledast}$.

∎

###  2.5. The triangular strategy

We have now all the ingredients to show the powerfulness of stated skein
algebras.

######  Strategy 2.21.

(Triangular strategy) Suppose you want to prove that (stated) skein algebras
satisfies a certain property (P).

  1. (1)

First prove (P) in the case of the bigon. Since
$\mathcal{S}_{A}(\mathcal{B})=\mathcal{O}_{q}[\operatorname{SL}_{2}]$ has been
overly studied, property (P) has probably already been proved in the
literature in this case (you might look at [Kas95, BG02b] for instance).

  2. (2)

Using the fact that
$\mathcal{S}_{A}(\mathbb{T})\cong\mathcal{O}_{q}[\operatorname{SL}_{2}]\circledast\mathcal{O}_{q}[\operatorname{SL}_{2}]$,
deduce that property (P) holds for the triangle.

  3. (3)

Using the embedding
$\theta^{\Delta}:\mathcal{S}_{A}(\mathbf{\Sigma})\hookrightarrow\otimes_{\mathbb{T}\in
F(\Delta)}\mathcal{S}_{A}(\mathbb{T})$ and the exact sequence (11), deduce
that property (P) is true for any triangulable surfaces.

  4. (4)

It remains to deal with non triangulable connected surfaces (other than the
bigon). Since
$\mathcal{S}_{A}(\mathbf{m}_{0})\cong\mathcal{S}_{A}(\mathbb{D}^{2},\emptyset)\cong
k$ and $\mathcal{S}_{A}(\Sigma_{0,1},\emptyset)\cong k[X]$, property (P) is
probably trivial in these cases, so it remains to deal with closed connected
(unmarked) surfaces $(\Sigma_{g,0},\emptyset)$. Since
$\mathbf{\Sigma}=(\Sigma_{g,0},\emptyset)$ is obtained from
$\mathbf{\Sigma}^{\prime}=(\Sigma_{g,1},\emptyset)$ by removing a puncture
(the unique boundary component), using the exact sequence

$0\to\mathcal{I}_{p}\to\mathcal{S}_{A}(\mathbf{\Sigma})\xrightarrow{f_{*}}\mathcal{S}_{A}(\mathbf{\Sigma}^{\prime})\to
0,$

and the fact (Proposition 2.7) that $\mathcal{I}_{p}$ is generated by elements
$[\gamma]-[\gamma^{\prime}]$ where $\gamma,\gamma^{\prime}$ are two closed
curves in $\Sigma_{g,1}$ which are isotopic in $\Sigma_{g,0}$, you can
probably deduce that property (P) holds for $\mathbf{\Sigma}$ from the fact
that it holds for the triangulable surface $\mathbf{\Sigma}^{\prime}$.

Let us illustrate this strategy on the following:

######  Theorem 2.22.

[BW11, PS19, Le18] If $k$ is a domain, so is
$\mathcal{S}_{A}(\mathbf{\Sigma})$.

Proving Theorem 2.22 without the use of stated skein algebras is a difficult
problem. In the particular case where $k=\mathbb{C}$, $A=-1$ and
$\mathbf{\Sigma}$ is closed, it was proved independently by Przytycki-Sikora
in [PS00] and Charles-Marche in [CM09] that the (commutative) skein algebra is
reduced.

The more general case of triangulable surfaces for any domain $k$ was proved
by Bonahon and Wong in [BW11] using the quantum trace (see also Müller [M1̈6]
for a similar strategy using another quantum torus). The case of closed
surfaces was proved by Przytycki-Sikora in [PS19] using suitable filtrations
from pants decomposition. Using the triangular strategy, Lê arrived at the
following trivial proof:

###### Proof.

(Lê [Le18]) The fact that $\mathcal{O}_{q}[\operatorname{SL}_{2}]$ is a domain
is classical and a proof (using a PBW basis) can be found in any textbook on
the subject like [Kas95]. It easily results that the cobraided tensor product
$\mathcal{O}_{q}[\operatorname{SL}_{2}]\circledast\mathcal{O}_{q}[\operatorname{SL}_{2}]$
is a domain. Since a subring of a domain is a domain, using the embedding
$\theta^{\Delta}:\mathcal{S}_{A}(\mathbf{\Sigma})\hookrightarrow\otimes_{\mathbb{T}\in
F(\Delta)}\mathcal{S}_{A}(\mathbb{T})$, we see that
$\mathcal{S}_{A}(\mathbf{\Sigma})$ is a domain whenever $\mathbf{\Sigma}$ is
triangulable.

∎

In this survey we will give three more illustrations of how the triangular
strategy can be used to get (almost) trivial proofs of deep results on skein
algebras. They are: $(1)$ the construction of the Chebyshev-Frobenius
morphism, $(2)$ the construction of the quantum trace (which was the original
motivation for the introduction of stated skein algebras) and $(3)$ the proof
that (stated) skein algebras are deformation quantizations of (relative)
character variety.

###  2.6. Chebyshev-Frobenius morphisms

######  Definition 2.23.

The $N$-th Chebyshev polynomial of first kind is the polynomial
$T_{N}(X)\in\mathbb{Z}[X]$ defined by the recursive formulas $T_{0}(X)=2$,
$T_{1}(X)=X$ and $T_{n+2}(X)=XT_{n+1}(X)-T_{n}(X)$ for $n\geq 0$.

The Chebyshev polynomial is alternatively defined by the equality

$\operatorname{tr}(A^{N})=T_{N}(\operatorname{tr}(A)),\quad\forall
A\in\operatorname{SL}_{2}(\mathbb{C}).$

It appears at the quantum level as follows. Suppose that $q^{2}=A^{4}$ is a
root of unity of order $N$. Then in $\mathcal{O}_{q}[\operatorname{SL}_{2}]$,
we have the following equality

(12) $T_{N}(\alpha_{++}+\alpha_{--})=\alpha_{++}^{N}+\alpha_{--}^{N}.$

Consider a stated arc $\alpha_{ij}$ in some marked surface $\mathbf{\Sigma}$
and denote by $\alpha_{ij}^{(N)}$ the stated tangle obtained by taking $N$
parallel copies of $\alpha_{ij}$ pushed along the framing direction. In the
case where both endpoints of $\alpha$ lye in two distinct boundary arcs, one
has the equality $\alpha_{ij}^{(N)}=(\alpha_{ij})^{N}$ in
$\mathcal{S}_{A}(\mathbf{\Sigma})$, but when both endpoints lye in the same
boundary arc, they are distinct. More precisely, suppose the two endpoints,
say $v$ and $w$, of $\alpha$ lye in the same boundary arc with $h(v)>h(w)$.
Then $\alpha_{ij}^{(N)}$ is defined by a stated tangle $(\alpha^{(N)},s)$
where $\alpha^{(N)}=\alpha_{1}\cup\ldots\cup\alpha_{N}$ represents $N$ copies