#  Planar $3$-webs and the boundary measurement matrix

Richard Kenyon111Department of Mathematics, Yale University, New Haven;
richard.kenyon at yale.edu Haolin Shi222Department of Mathematics, Yale
University, New Haven; haolin.shi at yale.edu

###### Abstract

We compute connection probabilities for reduced $3$-webs in the triple-dimer
model on circular planar graphs using the boundary measurement matrix (reduced
Kasteleyn matrix). As one application we compute several “$SL_{3}$
generalizations” of the Lindstrøm-Gessel-Viennot theorem, for “parallel” webs
and for honeycomb webs. We also apply our results to the scaling limit of the
dimer model in a planar domain, giving conformally invariant expressions for
reduced web probabilities.

##  1 Introduction

The dimer model is the study of the set of dimer covers, or perfect matchings,
of a planar graph. It is a highly successful combinatorial and probabilistic
model with connections to the Grassmannian (see e.g. [Pos]), conformal field
theory (see e.g. [Ken09]) and integrable systems [GK13].

A _double dimer cover_ of a graph is obtained by superimposing two dimer
covers, to get a collection of loops and doubled edges. In [KW11] Kenyon and
Wilson studied _boundary connection probabilities_ for the double dimer model
in planar graphs (see an example in Figure 1),

Figure 1: The union of two independent uniform dimer covers of a rectangle
(the first of which does not use the four corner vertices) produces a double
dimer configuration containing two chains of dimers connecting the four
corners in two possible ways. The probability of either connection is a
function of the boundary measurement matrix [KW11].

showing how they could be computed from the reduced Kasteleyn matrix, or
“boundary measurement matrix”. As application, in the scaling limit they
computed connection probabilities for multiple $\text{SLE}_{4}$ curves in the
disk. Similar calculations were recently done in [LPW] for the spanning tree
model and $\text{SLE}_{8}$ from a conformal field theory point of view. In
[Ken14, Dub19, BC21] it was shown how to use
$\text{SL}_{2}({\mathbb{R}})$-connections on graphs to compute various
topological connection probabilities for the double dimer model on a graph on
a topologically nontrivial surface without boundary.

Given these results, it is natural to ask about possible $n$-fold dimer
analogs, for $n\geq 3$. A union of $n$ dimer covers makes a more complicated
structure called a _graphical $n$-web_, or _$n$ -multiweb_, see [DKS22]. We
consider the case $n=3$ here, which has an extra feature not present in the
$n\geq 4$ case, which is the notion of _reduced_ webs.

Webs are primarily representation-theoretic objects. For $n=3$, Kuperberg in
[Kup96] used them to describe invariant functions in tensor products of
$\text{SL}_{3}$-representations. He also gave a diagrammatic basis for
invariant functions consisting of reduced $3$-webs. In [Pyl10], Pylyavskyy
discussed relations between reduced $3$-webs and the combinatorics of totally
positive matrices. Lam in [Lam15] first discusses the connection between
$3$-webs and the boundary measurement matrix, concentrating on the algebraic
and integrable aspects, in particular for positroid varieties. Fraser, Lam, Le
in [FLL19] also connect $n$-fold dimers with the boundary measurement matrix,
studying in particular the structure of the space of $n$-webs for $n\geq 4$.
In [DKS22] the connection was made between $\text{SL}_{n}$ web traces and the
associated Kasteleyn matrix determinant, for $n$-webs in general planar
bipartite graphs.

Even though they have a representation-theoretic underpinning, webs are also
natural from a combinatorial point of view: reduced $3$-webs are certain
topological types of configurations in the triple-dimer model on a bipartite
surface graph. Here we study (reduced) webs _as random objects_. We can
describe our main result as follows. To an edge-weighted planar graph with $n$
specified boundary vertices and specified boundary colors, we take a random
triple-dimer cover. The probability that a random reduction of this triple
dimer cover has a specified topological type is then a function of the
underlying weights. Our main result, Theorem 4, is an explicit expression for
this probability as a function of the reduced Kasteleyn matrix, or boundary
measurement matrix (sometimes referred to as _Postnikov’s boundary measurement
matrix_ since Postnikov first made use of it in his study of the totally
nonnegative Grassmannian [Pos]). Theorem 4 shows that the probability is a
constant times an integer-coefficient polynomial in the entries of the
boundary measurement matrix.

As an illustration of our results, for a planar graph with $3$ white and $3$
black boundary vertices in circular order $w,b,w,b,w,b$, the boundary
measurent matrix $X$ is a $3\times 3$ matrix and the probability of the web of
Figure 2

Figure 2: Example of a reduced web with $6$ nodes in order $w,b,w,b,w,b$.

(for “monochromatic” boundary conditions) is
$Pr=-2\frac{X_{12}X_{23}X_{31}}{\det X}.$ In the scaling limit of the triple
dimer model on the upper half plane, with $6$ marked boundary points as shown
in Figure 3, and the same monochromatic boundary conditions, the probability
of this reduced web as a function of the boundary points
$z_{1}<z_{2}<\dots<z_{6}\in{\mathbb{R}}$, is

$\text{Pr}=\frac{2(z_{2}-z_{1})(z_{3}-z_{2})(z_{4}-z_{3})(z_{5}-z_{4})(z_{6}-z_{5})(z_{6}-z_{1})}{(z_{3}-z_{1})(z_{4}-z_{2})(z_{5}-z_{3})(z_{6}-z_{4})(z_{5}-z_{1})(z_{6}-z_{2})}.$
(1)

The five other reduced webs are shown in Figure 30, and have probabilities
with similar expressions, see (28). See (27) for a case with $6n$ nodes.

Figure 3: A reduced web in the upper half plane with $6$ nodes.

As application of Theorem 4 we give several infinite families of webs whose
probabilities have explicit and compact expressions. In Section 6 we give an
“$\text{SL}_{3}$-version” of the Lindstrøm-Gessel-Viennot theorem, where we
compute the partition function, that is, unnormalized probability, for
parallel oriented crossings (see Figure 20) or parallel crossings with
“crossbars” (Figure 23) as a product of two determinants. In Section 7.1 we
show that the order-$n$ triangular honeycomb web $T_{n}$ of Figure 24 (of
which Figure 3 is the order-$1$ example) has a partition function which is a
product of three determinants. The result for another kind of order-$n$
triangular honeycomb (Figure 27) is given in Section 7.2.

In Section 8 we discuss the limiting probabilities of reduced webs for the
natural scaling limit of the square grid graph on the upper half plane or
other planar domains.

Here is a short summary of the main result (Theorem 4) and method of proof.
Given a circular planar bipartite graph ${\mathcal{G}}$ define the partition
function $Z:=\sum_{\text{multiwebs $m$}}\mathrm{Tr}_{m}$. Each
$\mathrm{Tr}_{m}$ can be expanded in terms of the basis
$\\{\mathrm{Tr}_{\lambda}\\}_{\lambda\in\Lambda}$ of reduced webs, leading to
$Z=\sum_{\lambda\in\Lambda}C_{\lambda}\mathrm{Tr}_{\lambda},$ for some
coefficients $C_{\lambda}$. The goal is to compute $C_{\lambda}$. For each
node coloring $c$, we can compute $Z(c)$ as a product of three determinants,
and expand these using an “intermediate” spanning set of functions
$\\{\mathrm{Tr}_{\tau}\\}_{\tau\in\Pi}$ where $\tau$ runs over a simple family
of nonplanar webs. We can then rewrite each $\mathrm{Tr}_{\tau}$ in terms of
the reduced webs $\mathrm{Tr}_{\lambda}$, and thus write
$\mathrm{Tr}_{\tau}(c)$ in terms of $\mathrm{Tr}_{\lambda}(c)$, in such a way
that the coefficients $C_{\lambda}$ do not depend on $c$. Getting the signs
right is the hardest part of this theory; the key argument is Lemma 2.

Acknowledgments. This research was supported by NSF grant DMS-1940932 and the
Simons Foundation grant 327929. We thank Daniel Douglas, Pavlo Pylyavskyy and
Xin Sun for discussions and insights.

##  2 Definitions and background

###  2.1 The graph

Let ${\mathcal{G}}=(V,E)$ be a circular planar bipartite graph. This means
${\mathcal{G}}$ is a bipartite graph: $V=B\cup W$ with $E\subset B\times W$,
and ${\mathcal{G}}$ is embedded in a disk with a specified set of vertices
$V_{\partial}\subset V$ on the bounding circle; $V_{\partial}$ contains some
but not necessarily all vertices on the outer face of ${\mathcal{G}}$. We
refer to vertices in $V_{\partial}$ as _nodes_ , and vertices in $V\setminus
V_{\partial}$ as _internal vertices_.

We denote by $W_{\partial},B_{\partial}$ the white and black nodes, and by
$W_{int},B_{int}$ the white and black internal vertices. We assume throughout
the paper that $|W_{\partial}|\geq|B_{\partial}|$. The case
$|W_{\partial}|\leq|B_{\partial}|$ is equivalent after exchanging colors. We
will also assume

$3(|B_{int}|-|W_{int}|)=|W_{\partial}|-|B_{\partial}|,$

see (3), below.

We assume ${\mathcal{G}}$ is nondegenerate in the following (mild) technical
sense. The graph ${\mathcal{G}}$ is required to have two kinds of partial
matchings; see Figure 4 for an illustration. In the case that
$|W_{\partial}|=|B_{\partial}|$, we require simply that ${\mathcal{G}}$ have a
partial dimer cover covering exactly the internal vertices, and also that
${\mathcal{G}}$ have a full dimer cover covering all vertices. More generally,
if $|W_{\partial}|\geq|B_{\partial}|$, then we require first, that
${\mathcal{G}}$ have a partial dimer cover $M$ which covers all internal
vertices and no black node (and therefore some set of white nodes $W^{*}$ if
$|W_{\partial}|>|B_{\partial}|$), see Figure 4, left. We let
$B_{int}^{*}\subset B_{int}$ denote the internal black vertices connected in
$M$ to $W^{*}$. Necessarily $|W^{*}|=|B_{int}^{*}|=|B_{int}|-|W_{int}|$. The
second required partial dimer cover $M^{\prime}$ of ${\mathcal{G}}$ uses all
vertices except $W^{*}\cup W^{**}$ where $W^{*}$ is defined from $M$ as above
and $W^{**}$ consists in, for each $w\in W^{*}$, one white vertex (node or
internal) $w^{\prime}\neq w$ on one of the two external faces adjacent to $w$.
(By external face we mean a complementary component of ${\mathcal{G}}$ in the
disk, adjacent to the boundary of the disk.) See Figure 4, right. The
existence of the first kind of partial matching is essential while the
existence of the second is probably not essential but makes our proof easier
(as it allows us to reduce the $|W_{\partial}|>|B_{\partial}|$ case to the
$|W_{\partial}|=|B_{\partial}|$ case…. see Figure 19).

Figure 4: For nondegeneracy, we need two partial matchings. One, on the left,
is a partial matching $M$ using all internal vertices and no black node. This
has $B_{int}^{*}=\\{b_{4},b_{5}\\}$ and $W^{*}=\\{w_{2},w_{4}\\}$. The second,
on the right, is a partial matching $M^{\prime}$ using all vertices except
$W^{*}=\\{w_{2},w_{4}\\}$ and $W^{**}=\\{w_{1},w_{9}\\}$. Note that $w_{1}$ is
on the same external face as $w_{2}$ and $w_{9}$ is on the same external face
as $w_{4}$.

We let $\nu:E\to{\mathbb{R}}_{>0}$ be a positive weight function on the edges
of ${\mathcal{G}}$.

###  2.2 Webs, skein relations and reduced web classes

A _multiweb_ in ${\mathcal{G}}$ is a map $m:E\to\\{0,1,2,3\\}$ with the
property that for internal vertices $v$, $\sum_{u\sim v}m_{uv}=3$ and for
boundary vertices $v$, $\sum_{u\sim v}m_{uv}=1.$ See Figure 5. In other words
a multiweb is a submultigraph with degree $3$ at internal vertices and degree
$1$ at each node $v$.

Figure 5: Left: example of a multiweb (blue) $m$ in a $5\times 4$ grid graph.
Multiplicities of edges are indicated by their thickness. Nodes are at the
corners (blue dots). Right: the associated abstract web $[m]$.

The _weight_ $\nu(m)$ of a multiweb $m$ is defined to be the product of its
edge weights, taking into account their multiplicity:

$\nu(m)=\prod_{e\in E}\nu_{e}^{m_{e}}.$

If we ignore the edges of multiplicity $3$, which are isolated, a multiweb
consists of a set of trivalent vertices (vertices with three distinct edges of
multiplicity $1$) and the boundary vertices, connected via “$12$-paths”, that
is, paths of edges which are alternately of multiplicity $1$ and $2$,
beginning and ending with an edge of multiplicity $1$ (these paths may consist
in a single edge of multiplicity $1$). There may be also some closed loops of
$12$-paths. A multiweb $m$ has an associated _web_ $[m]$, which is an abstract
unweighted graph with vertices in bijection with the union of the boundary
vertices and trivalent vertices of $m$, and two vertices are connected in
$[m]$ if they are connected in $m$ with a $12$-path. In particular the
multiplicity-$3$ edges of $m$ are ignored in $[m]$. The abstract web $[m]$
also has disjoint loops (with no vertices) for each closed $12$-path in $m$.
Note that $[m]$ is a circular planar bipartite graph (possibly with loops)
with boundary $V_{\partial}$. See Figure 5.

A web $[m]$ is _reduced_ if it has no faces of degree $0,2$ or $4$. We say a
multiweb $m$ is reduced if its associated web $[m]$ is reduced. Note that in a
reduced web, every component contains at least one node: a component without a
node would be a trivalent bipartite planar graph with no faces of degree $2$
or $4$. A simple Euler characteristic argument shows that such graphs do not
exist. Thus a multiweb is reduced if the connected components of its
complement are bounded by paths of edges which contain either a node or at
least $6$ trivalent vertices.

To a multiweb $m$ is associated a function, the _trace_ $\mathrm{Tr}_{m}$,
defined in Section 2.5 below. It is a multilinear function of a set of vectors
in ${\mathbb{R}}^{3}$ assigned to the nodes, one to each node. The trace only
depends on the underlying abstract web: $\mathrm{Tr}_{m}=\mathrm{Tr}_{[m]}$.
As shown in [Kup96]333We use the sign convention of Sikora [Sik01] which
differs from that of Kuperberg., any planar web can be replaced by a formal
linear combination of reduced webs by applying a sequence of _skein relations_
, see Figure 6. These skein relations preserve the trace, in the sense that
the trace of a web equals the sum of the traces of the webs in its reduction.

Figure 6: Planar skein reductions for webs. First, any closed loop can be
removed, multiplying the coefficient of the remaining web by $3$. Any bigon
can be removed as shown, doubling the coefficient. Thirdly, any square face
can be replaced by two parallel paths in two ways, as shown.  Figure 7: The
basic skein relation, with Sikora sign convention.

The second and third skein relations follow from the more basic relation of
Figure 7, which involves webs with edge crossings. We can use the basic skein
relation to replace any nonplanar web (that is, web drawn on the plane with
edge crossings) by a formal linear combination of planar webs, by resolving
each crossing locally into the other two locally planar pieces. It is worth
noting that if two strands of a web cross more than once consecutively,
applying skein relations to both these crossings reduces the web to the same
web but with both crossings removed, as in the second Reidemeister move.
Likewise if three strands mutually cross then applying a third Reidemeister
move results in an equivalent web (equivalent under skein relations). Finally,
if a strand crosses itself then it is equivalent to the uncrossed version as
in the first Reidemeister move. These observations show that when dealing with
webs having edge crossings we are free to isotope the strands across other
strands and/or crossings as desired.

The skein relations have analogs for multiwebs as well, see for example Figure
8. For a bigon or quad face, these relations involve changing the multiplicity
of edges on the face by $\pm 1$ alternatively around the face. That is, if
multiplicities are $m_{1},m_{2},\dots,m_{2k}$ then replace these with
multiplicities $m_{1}+1,m_{2}-1,\dots,m_{2k}-1$ to get one multiweb and by
$m_{1}-1,m_{2}+1,\dots,m_{2k}+1$ to get the other multiweb. Both resulting
multiwebs have one fewer face.

Figure 8: Example of the second skein relation realized as a multiweb skein
relation. Note that it replaces a multiweb having a bigon face with two
different (but equivalent) multiwebs without the bigon face.

A _reduced web class_ is an equivalence class of reduced multiwebs, two
reduced multiwebs being equivalent if their abstract (reduced) webs are
isomorphic with an isomorphism preserving the nodes pointwise.

Let $\Lambda$ be the set of reduced web classes arising in ${\mathcal{G}}$.
See Figures 13, 15, 17, 30, 31, 32 and 33 for examples. Given a multiweb $m$
in ${\mathcal{G}}$, upon applying skein relations we can write $m$ as a formal
integer linear combination of reduced web classes:

$m=\sum_{\lambda\in\Lambda}C_{\lambda}(m)\lambda$ (2)

where the $C_{\lambda}(m)$ are nonnegative integers. See Figure 9 for an
example.

Figure 9: Example of reducing a web with four nodes. We first apply a skein
relation to the left quadrilateral face, then apply the skein relation to the
resulting bigonal face.  Figure 10: The same reduction as in Figure 9 using
multiweb skein relations. Of the three resulting reduced multiwebs, the first
two have the same web class.

###  2.3 Node types

To not confuse the notions of color of a vertex (black or white) with the
color of an edge, we refer to the blackness or whiteness of a vertex as its
_type_.

Fix ${\mathcal{G}}$ with $n$ nodes $V_{\partial}=\\{v_{1},\dots,v_{n}\\}$ in
counterclockwise order. We refer to the $n$-tuple $p\in\\{b,w\\}^{n}$ of types
as the _type vector_ of ${\mathcal{G}}$. Let $\Omega$ be the set of multiwebs
in ${\mathcal{G}}$, with multiplicities $1$ on $V_{\partial}$. For $\Omega$ to
be nonempty there is a linear constraint on the number of black and white
internal vertices and black and white nodes: we must have

$3(|B_{int}|-|W_{int}|)=|W_{\partial}|-|B_{\partial}|.$ (3)

This follows by bipartiteness: $3|W_{int}|+|W_{\partial}|$ is the total white
degree, which must equal the total black degree, since each edge has one black
and one white vertex. The quantity $|B_{int}|-|W_{int}|$ is sometimes called
the _excedence_ in dimer literature.

From (3), $|W_{\partial}|-|B_{\partial}|$ must be a multiple of $3$. For
example, for webs with $2,3,4$ or $5$ nodes the possible total types
$(|W_{\partial}|,|B_{\partial}|)$ must be respectively
$(1,1),(3,0),(2,2),(4,1)$, and for $6$ nodes there are two possible total
types, $(3,3)$ or $(6,0)$ (recall that we make the standing assumption that
$|W_{\partial}|\geq|B_{\partial}|$).

The maximal number of reduced web classes for ${\mathcal{G}}$ with
$(|W_{\partial}|,|B_{\partial}|)$ is discussed in Section 2.6 below.

###  2.4 Colored multiwebs

Let ${\mathcal{C}}=\\{1,2,3\\}$ be the fixed set of colors. An _edge coloring_
of a web $m$ is an assignment of a color in ${\mathcal{C}}$ to each edge so
that at each trivalent vertex all three colors are present. The definition of
edge coloring is slightly different for multiwebs, to take into account the
multiplicity of edges. A _colored multiweb_ is a multiweb in which each edge
of multiplicity $k$ has an associated subset $S_{e}\subset{\mathcal{C}}$ of
size $k$ ($S_{e}$ is the _set of colors of edge $e$_), with the property that
at each internal vertex $v$ all colors are present: $\cup_{u\sim
v}S_{uv}={\mathcal{C}}$. At a node (of degree $1$) only one color will be
present. See an example in Figure 11.

Figure 11: Coloring of a multiweb, where the lower two vertices are the nodes.

Given a choice $c:V_{\partial}\to{\mathcal{C}}$ of colors at the nodes, define
$\Omega_{c}$ to be the set of colored multiwebs in ${\mathcal{G}}$ in which
the unique edge at node $v$ has color $c(v)$. We say $c$ is _feasible_ if
$\Omega_{c}$ is nonempty; $\Omega_{c}$ is nonempty only when $c$ satisfies
certain constraints. The set of edges of each color in a colored multiweb is a
dimer cover of the graph ${\mathcal{G}}_{i}$ obtained from ${\mathcal{G}}$ by
removing nodes not in $V_{i}$. So $\Omega_{c}$ is nonempty if and only if each
${\mathcal{G}}_{i}$ has a dimer cover.

In particular the number of white nodes of color $i$, plus the number of white
internal vertices, must equal the number of black nodes of color $i$ plus the
number of black internal vertices:

$|W_{i}|-|B_{i}|=|B_{int}|-|W_{int}|.$ (4)

These equations (4) give a necessary, but not generally sufficient, criterion
for $\Omega_{c}$ to be nonempty.

###  2.5 Traces

Let $Y$ be a $3$-dimensional ${\mathbb{R}}$-vector space,
$Y\cong{\mathbb{R}}^{3}$, and $Y^{*}=\text{Hom}(Y,{\mathbb{R}})$ its dual.
Associated to a web $[m]$ with $(|W_{\partial}|,|B_{\partial}|)=(k,n-k)$ is an
$\text{SL}_{3}({\mathbb{R}})$-invariant multilinear function, its _trace_ ,
$\mathrm{Tr}_{[m]}:Y^{\otimes k}\otimes(Y^{*})^{\otimes n-k}\to{\mathbb{R}}$.
That is, $\mathrm{Tr}_{[m]}$ is a real valued, multilinear function of $k$
vectors in $Y$ and $n-k$ covectors in $Y^{*}$, which is invariant under the
natural diagonal action of $\text{SL}_{3}$. For the tensorial definition of
$\mathrm{Tr}_{[m]}$ see [Kup96] or [DKS22]. We give here a concrete
combinatorial definition, which is equivalent (see [DKS22]) to the tensorial
one, although it has the disadvantage that the $\text{SL}_{3}$-invariance is
not obvious.

Let $e_{1},e_{2},e_{3}\in Y$ be a basis in $Y$ and
$e_{1}^{*},e_{2}^{*},e_{3}^{*}\in Y^{*}$ a dual basis. We associate color
$i\in{\mathcal{C}}$ to $e_{i}$ and $e_{i}^{*}$. By multilinearity, it suffices
to define $\mathrm{Tr}_{[m]}$ on $k$-tuples of basis vectors in $Y$ and
$(n-k)$-tuples of basis covectors in $Y^{*}$.

By definition, the result of applying $\mathrm{Tr}_{[m]}$ to
$\vec{e}=(e_{i_{1}},\dots,e_{i_{k}},e_{j_{1}}^{*},\dots,e_{j_{n-k}}^{*})$ is
an integer equal to a signed number of edge colorings of $[m]$ in which the
$\ell$th white node has its unique edge of color $i_{\ell}$ and the $\ell$th
black node has its unique edge of color $j_{\ell}$. The signs are defined as
follows. Given an edge coloring $\eta$ of $[m]$, at each interior vertex the
colors are either in counterclockwise order or clockwise order. At each
interior black vertex define the local sign to be $1$ if the order is
counterclockwise and $-1$ if clockwise; reverse these conventions at interior
white vertices. The sign of a coloring is by definition the product over all
interior vertices of these local signs, and $\mathrm{Tr}_{[m]}(\vec{e})$ is
the signed number of colored webs whose underlying web is $[m]$.

See Figure 12 for the most basic examples. The “tripod” web $[m]$ shown with
$(|W_{\partial}|,|B_{\partial}|)=(3,0)$ has
$\mathrm{Tr}_{[m]}(u,v,w)=\det(u,v,w)$, the determinant of the matrix with
columns $u,v,w$. Note that the determinant is multilinear and
$\text{SL}_{3}$-invariant: $\det(u,v,w)=\det(Au,Av,Aw)$ for any
$A\in\text{SL}_{3}$. Moreover this web has exactly $6$ edge colorings, $3$ of
each sign, which correspond to the six terms in the expansion (writing $u,v,w$
in the basis $e_{1},e_{2},e_{3}$):

$\det(u,v,w)=u_{1}v_{2}w_{3}-u_{1}v_{3}w_{2}+u_{2}v_{3}w_{1}-u_{2}v_{1}w_{3}+u_{3}v_{1}w_{2}-u_{3}v_{2}w_{1}.$

The “line” web $[m]$ shown with $(|W_{\partial}|,|B_{\partial}|)=(1,1)$ has
$\mathrm{Tr}_{[m]}(u,v^{*})=v^{*}(u)$. This is a multilinear function of $u$
and $v^{*}$, and also $\text{SL}_{3}$ invariant (since we _define_ the action
of $\text{SL}_{3}$ on $Y^{*}$ in such a way as to preserve the pairing). This
web has exactly $3$ edge colorings, which correspond to the three terms in the
expansion

$v^{*}(u)=v_{1}u_{1}+v_{2}u_{2}+v_{3}u_{3}.$

Figure 12: Left: the “tripod” web $[m]$ shown has $\mathrm{Tr}_{[m]}(u,v,w)$
equal to the determinant $u\wedge v\wedge w$. Right: the “line” web $[m]$ has
$\mathrm{Tr}_{[m]}(u,v^{*})=v^{*}(u)$.

The trace for multiwebs is defined in the same way as the trace for webs,
where we only take into account the trivalent internal vertices when computing
the sign. We have $\mathrm{Tr}_{m}=\mathrm{Tr}_{[m]}$.

In [DKS22] it was shown that for a planar graph _without_ boundary, all
colorings of a web or multiweb have positive sign, so the signed number of
colorings equals the number of colorings. A similar result holds for webs with
fixed boundary colors.

######  Lemma 1.

For a circular planar graph with choice of node colors $c$, all colorings of
all multiwebs have the same sign $\varepsilon_{c}$.

###### Proof.

We embed ${\mathcal{G}}$ in a larger planar graph ${\mathcal{G}}^{\prime}$ by
adding edges outside the bounding disk, in such a way that any colored
multiweb $m$ in ${\mathcal{G}}$ with node colors $c$ extends to a colored
multiweb $m^{\prime}$ without boundary in ${\mathcal{G}}^{\prime}$. Fix the
part of such a multiweb $m^{\prime}$ outside of ${\mathcal{G}}$. Then the sign
of $m^{\prime}$, which is positive by [DKS22], is the product of the sign of
$m$ and the sign of the part of $m^{\prime}$ outside of ${\mathcal{G}}$, which
is fixed. ∎

One method of computing the sign $\varepsilon_{c}$ directly is below in
Section 2.6.

For a reduced multiweb $m$, the trace only depends on the web class
$[m]\in\Lambda$. Skein relations preserve the trace, in the sense that the
trace of a web or multiweb equals the sum of the traces of the webs resulting
from it in applying a skein relation. Applying the trace to both sides of (2)
thus gives

$\mathrm{Tr}_{m}=\sum_{\lambda\in\Lambda}C_{\lambda}(m)\mathrm{Tr}_{\lambda}.$
(5)

Kuperberg showed in [Kup96] that the
$\\{\mathrm{Tr}_{\lambda}\\}_{\lambda\in\Lambda}$ are linearly independent on
$(Y^{\otimes n_{1}}\otimes(Y^{*})^{\otimes n_{2}})^{\text{SL}_{3}}$, that is
on the space of $\text{SL}_{3}$-invariant functions on $Y^{\otimes
n_{1}}\otimes(Y^{*})^{\otimes n_{2}}$. Thus the coefficients $C_{\lambda}(m)$
in (5) are well defined functions of $m$.

###  2.6 $3$-partitions

Let ${\mathcal{G}}=(V,E)$ with $|V_{\partial}|=n$, with
$V_{\partial}=W_{\partial}\cup B_{\partial}$ as above,
$|W_{\partial}|\geq|B_{\partial}|$, and let ${\mathcal{G}}$ have type vector
$p$. Let $\Pi$ be the set of unordered, not-necessarily-planar partitions of
the nodes into black/white pairs and white triples. Let us refer to such
partitions as “$3$-partitions”. For a $3$-partition, the number of white
triples of nodes is exactly
$N:=\frac{1}{3}(W_{\partial}-B_{\partial})=B_{int}-W_{int}$ (by (3)).

Letting $k=|W_{\partial}|$, the number of elements of $\Pi$ is
$\frac{k!}{N!6^{N}}$. This is because there are $\frac{k!}{(2k-n)!}$
injections from $B_{\partial}$ (of size $n-k$) to $W_{\partial}$ (of size
$k$), and then $\frac{(3N)!}{N!(3!)^{N}}$ ways of partitioning the remaining
$3N$ white nodes into triples (and $3N=2k-n$).

The number of reduced web classes was shown in [PPR09] to not depend on $p$,
only on $(|W_{\partial}|,|B_{\partial}|)$, and to be the Kostka number
$K_{3^{m},2^{k}1^{n-k}}$ where $k=|W_{\partial}|$ and $m=(n+k)/3$. The Kostka
number $K_{\mu,\nu}$ is the number of semistandard Young tableaux (SSYT) of
shape $\mu$ and weight $\nu$. In this case, it is the number of ways to fill a
$3\times m$ table ($3$ columns, $m$ rows) with numbers
$1,1,2,2,\dots,k,k,k+1,k+2,\dots,n$ so that numbers increase along columns and
do not decrease along rows. For example when
$(|W_{\partial}|,|B_{\partial}|)=(4,1)$, as in Figure 17, we have $n=5,k=4$
and $m=3$. The set of SSYT is

4 4 5 2 3 3 1 1 2 3 4 5 2 3 4 1 1 2 3 4 5 2 2 4 1 1 3

and the Kosta number is $K_{3^{3},2^{4}1^{1}}=3$.

We can associate to a $3$-partition a crossing diagram in the disk, by drawing
a chord connecting pairs of the $3$-partition and, for every triple
$t=w_{1}w_{2}w_{3}$, drawing a “tripod”: putting an interior black vertex
$b=b_{t}$ in the disk and connecting it to each of $w_{1},w_{2},w_{3}$ with a
line segment.

The trace of a $3$-partition $\tau$ is defined using the above drawing as for
the trace of a web; the nonplanarity plays no role in the definition of trace:
if $\tau\in\Pi$ and $c$ is a compatible coloring of the nodes, then
$\mathrm{Tr}_{\tau}(c)=(-1)^{\ell}$ where $\ell$ is the number of triples of
$\tau$ whose colors are in clockwise order. In particular if there are no
triples, $\mathrm{Tr}_{\tau}(c)=1$.

One way to compute the sign of $\varepsilon_{c}$ from Lemma 1 is as follows.
Find a $3$-partition $\tau$ of the nodes of ${\mathcal{G}}$. Draw $\tau$ in
the disk (with possibly crossing edges) by joining each triple of $\tau$ with
a black vertex $b$ in the disk, and drawing each pair of $\tau$ as a path
connecting its endpoints. Each path in $\tau$ inherits a color from its
boundary vertex/vertices. Let $T$ be the number of triples with colors in
clockwise order and let $M$ be the number of nonmonochromatic crossings
(NMCs): crossings of edges of different colors. Note that while the number of
NMCs depends on the choice of drawing, its parity is an isotopy invariant.
Then $\varepsilon_{c}=(-1)^{M+T}$. This follows because, using the basic skein
relation (see Figure 7) to make each crossing planar, each NMC resolves into a
double-Y with one vertex of each sign, and each monochromatic crossing
resolves into a pair of parallel edges (with no sign change). We have
constructed a colored planar web with sign $\varepsilon_{c}$, and thus by
Lemma 1 all planar webs with boundary colors $c$ have this same sign
$\varepsilon_{c}$.

###  2.7 Kasteleyn matrix and boundary measurement matrix

Associated to ${\mathcal{G}}$ and $\nu$ is a _Kasteleyn matrix_ $K$, a matrix
with rows indexing the white vertices and columns indexing the black vertices.
This is a signed, weighted adjacency matrix: $K_{wb}=0$ if $w$ is not adjacent
to $b$ and, if they are adjacent, $K_{wb}=\varepsilon_{wb}\nu(wb)$ where signs
$\varepsilon_{wb}\in\\{1,-1\\}$ are chosen so that a bounded face of length
$\ell$ has $\frac{\ell}{2}+1\bmod 2$ minus signs. See e.g. [Kas61, Ken09].
Different choices of such signs give to matrices related by “gauge
transformation”, that is, by left- and right-multiplication by diagonal
matrices of diagonal entries $\\{\pm 1\\}$.

We define $X$, the _reduced Kasteleyn matrix_ , or _boundary measurement
matrix_ , to be a maximal Schur reduction of $K$ to the nodes, in the
following sense. Take a partial matching $M$ which uses all internal vertices
and no black node. Such a matching exists by nondegeneracy, see Figure 4,
left. Let $B_{int}^{*}\subset B_{int}$ denote the internal black vertices
connected in $M$ to a white node. Let $D$ be the matrix $K$ restricted to the
vertices $W_{int}$ and $B_{int}\setminus B_{int}^{*}$: this is the set of
vertices of $M$ when we remove from $M$ the dimers connected to white nodes.
Then $D$ is a Kasteleyn matrix of its induced subgraph: each bounded face of
the subgraph is a face of ${\mathcal{G}}$. By ordering vertices of
${\mathcal{G}}$ so that $D$’s vertices are last, we can write
$K=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ in block form. Then define

$X:=A-BD^{-1}C.$ (6)

Let $\Delta=\det D$; this is nonzero since $D$ is a Kasteleyn matrix of a
subgraph of ${\mathcal{G}}$ which has at least one dimer cover.

Up to sign each entry $X_{wb}$ has the following combinatorial interpretation
[KW11]: $X_{wb}=\pm Z({\mathcal{G}}_{wb})/\Delta$ where
$Z({\mathcal{G}}_{wb})$ is the weighted sum of dimer covers of the graph
${\mathcal{G}}_{wb}$, the graph obtained from ${\mathcal{G}}$ by removing all
nodes and all of $B_{int^{*}}$ except $w$ and $b$. To see this, note that
removing rows and columns of $K$ for all nodes and vertices $B_{int}^{*}$ not
equal to $w$ or $b$ results in a matrix $\begin{pmatrix}A_{wb}&B_{w}\\\
C_{b}&D\end{pmatrix}$ whose Schur reduction is the $1\times 1$ matrix
$X_{wb}$, and whose determinant is $Z({\mathcal{G}}_{wb})$.

Note that $X$ depends nontrivially on our choice of $B_{int}^{*}$
(corresponding to submatrix $D$).

If $|W_{\partial}|=|B_{\partial}|$ then $K$ and $X$ are square matrices, and

$\det K=\det D\det X=\Delta\det X;$ (7)

this follows from applying the determinant to both sides of

$\begin{pmatrix}A&B\\\ C&D\end{pmatrix}=\begin{pmatrix}A-BD^{-1}C&B\\\
0&D\end{pmatrix}\begin{pmatrix}I&0\\\ D^{-1}C&I\end{pmatrix}.$

Note that $\det K\neq 0$ by nondegeneracy.

###  2.8 Signs

If $|W_{\partial}|=|B_{\partial}|=k$, then $K$ is a square $k\times k$ matrix.
Suppose $W_{1}\cup W_{2}\cup W_{3}$ is a partition of the white nodes and
$B_{1}\cup B_{2}\cup B_{3}$ is a partition of the black nodes, with
$|W_{i}|=|B_{i}|$ for each $i$. Let $c$ denote this partition; we think of $c$
as a coloring of the nodes with colors ${\mathcal{C}}=\\{1,2,3\\}$. Let
$K_{i}=K_{W_{i}}^{B_{i}}$ be the submatrix of $K$ obtained by discarding nodes
not in $W_{i}\cup B_{i}$, that is, keeping only interior vertices and nodes of
color $i$. Likewise define $X_{i}=X_{W_{i}}^{B_{i}}$, its Schur reduction
using the same submatrix $D$. We have $\Delta\det X_{i}=\det K_{i}$ (and
$\Delta\det X=\det K$, see above).

Let $\delta_{c}$ be the sign of the quantity $\det K_{1}\det K_{2}\det
K_{3}/(\det K)^{3}$, which is also the sign of $\det X_{1}\det X_{2}\det
X_{3}/(\det X)^{3}$ and the sign of $\det X_{1}\det X_{2}\det X_{3}/\det X$.
We define another sign, $\eta_{c}$, which is the sign of $\det X_{1}\det
X_{2}\det X_{3}$ as a polynomial in the entries of $X$ occurring in the
expansion of $\det X$.

It is convenient to choose an ordering for the nodes, which determines an
ordering for the rows and columns of $X$. Index the nodes of ${\mathcal{G}}$
counterclockwise for white vertices and clockwise for black vertices. Letting
$w_{1}$ be the first white node, we assume that $b_{1}$ is the first black
node clockwise from it. This particular ordering is just a convention, and
helps with the proof of Theorem 4 (although it does not affect the statement).
We take a different order in Section 7.1 and the examples in the appendix.

######  Lemma 2.

We have $\varepsilon_{c}=\delta_{c}\eta_{c}$.

###### Proof.

Consider all the vertices on the outer face of ${\mathcal{G}}$; some subset of
these are nodes. As noted above we index the nodes counterclockwise for white
vertices and clockwise for black vertices. Let $w_{1}$ be the first white
node, and assume that $b_{1}$ is the first black node clockwise from it.

Let us compute the sign of $\det X_{1}/\det X=\det K_{1}/\det K$. We add
external edges to ${\mathcal{G}}$ outside the disk, connecting white nodes
$W_{2}\cup W_{3}$ to black nodes $B_{2}\cup B_{3}$ in pairs, and so that the
resulting graph ${\mathcal{G}}^{1}$ is still planar. We can extend the
Kasteleyn signs on ${\mathcal{G}}$ to ${\mathcal{G}}^{1}$ by assigning signs
to the new edges. Let $K^{1}$ be the enlarged Kasteleyn matrix. In $\det
K^{1}$, all dimer covers have the same sign by Kasteleyn’s theorem; but dimer
covers which use none of the new edges have the same sign as in $\det K$. This
sign is also the sign of dimer covers using all the new edges, which by
definition have the sign of $\det K_{1}$ times the product of the new signs.
Thus $\det K_{1}/\det K$ has sign given by the product of the Kasteleyn signs
on the new edges. Likewise construct ${\mathcal{G}}^{2},{\mathcal{G}}^{3}$ and
$\det K_{2}/\det K,\det K_{3}/\det K.$

By the gauge invariance of the choice of Kasteleyn signs, we can choose
Kasteleyn signs on the original graph ${\mathcal{G}}$ so that all boundary
edges have sign $+$, except possibly for the edge $e_{0}$ just clockwise of
$w_{1}$ which has sign $-1$ if the boundary has length $0\bmod 4$. Then the
Kasteleyn sign on an external edge of ${\mathcal{G}}^{i}$ depends on the
distance $\ell$ in ${\mathcal{G}}$ between its vertices, measured around the
arc of the boundary which does not cross $e_{0}$: the sign is
$(-1)^{(\ell-1)/2}$.

Thus the sign of each $\det X_{i}/\det X$ depends only on the distances (mod
$4$) between nodes.

Suppose we move a black node $b_{i}$, of color $1$, increasing its distance
from the previous node by $2$ and decreasing its distance from the next node
by $2$. That is, we make $b_{i}$ an internal vertex and instead assign the
next (clockwise) black vertex around the boundary to be a node, the “new”
$b_{i}$. Then both $\det X_{2}/\det X$ and $\det X_{3}/\det X$ will change
sign (since whatever white node $b_{i}$ was matched to externally will have
its distance to $b_{i}$ changed by $\pm 2$), while $\det X_{1}/\det X$ will
not change sign. This results in no net sign change of $\delta_{c}$. Moreover
$\varepsilon_{c},\eta_{c}$ do not change under this operation. So we see that
$\delta_{c},\varepsilon_{c}$ and $\eta_{c}$ only depend on the circular order
of the nodes (and their colors), not their individual distances.

Suppose initially that white nodes and black nodes lie in disjoint intervals,
so that counterclockwise starting from $w_{1}$ we see
$w_{1},\dots,w_{n},b_{n},\dots,b_{1}$. Suppose also that colors are in the
counterclockwise order $W_{1},W_{2},W_{3}$ then $B_{3},B_{2},B_{1}$. That is,
starting from the first white node $w_{1}$ and proceeding counterclockwise we
see all white nodes of color $1$, then all white nodes of color $2$, and so
on, ending with all black nodes of color $1$. In this case we can add external
edges in each of ${\mathcal{G}}^{1},{\mathcal{G}}^{2},{\mathcal{G}}^{3}$
connecting each $w_{i}$ to $b_{i}$. In $\det X_{1}\det X_{2}\det X_{3}/(\det
X)^{3}$ each sign on these external edges contributes twice, so
$\delta_{c}=1$. Moreover $\varepsilon_{c}=1$ and $\eta_{c}=1$ in this case, so
the statement holds.

It remains to see how the sign changes when we reorder the boundary nodes or
colors. There are three cases to consider: we switch colors on adjacent nodes,
both of which are black or both of which are white; we switch positions of a
black and white adjacent node of the same color; and we switch positions of a
black and white adjacent node of different colors. In each case we compute the
change in sign of $\delta_{c},\eta_{c}$ and $\varepsilon_{c}$ and show that
$\varepsilon_{c}\delta_{c}\eta_{c}$ remains constant.

Suppose we switch colors, say color $1$ and $2$, on two adjacent black nodes,
separated by (even) distance $\ell$. Then both $\varepsilon_{c}$ and
$\eta_{c}$ will change. In both $\det X_{1}/\det X$ and $\det X_{2}/\det X$
the sign will change by $(-1)^{\ell/2}$, while the sign of $\det X_{3}/\det X$
will not change. Thus $\delta_{c}$ changes by $(-1)^{\ell}=+1$, that is, does
not change. A similar argument holds if we switch colors on two adjacent white
nodes.

Suppose we switch positions of two adjacent nodes $b,w$, one black and one
white, of the same color, say color $1$. Then $\eta_{c}$ and $\varepsilon_{c}$
do not change. In the graphs ${\mathcal{G}}^{2}$ and ${\mathcal{G}}^{3}$, we
can assume $b,w$ are paired by an external edge, since they are adjacent. When
we swap them, keeping the same distance, the signs of $\det X_{2}/\det X$ and
$\det X_{3}/\det X$ do not change; neither does the sign of $\det X_{1}/\det
X$. Thus $\delta_{c}$ does not change.

Suppose we switch positions of two adjacent nodes $b,w$, one black and one
white, of odd distance $\ell$, and of different colors, say colors $1$ and $2$
without loss of generality (and we think of colors as attached to the nodes,
so the colors also swap positions). Suppose for example $b$ advances by
$\ell+1$ cclw around the boundary, and $w$ retreats by $\ell-1$. Then
$\varepsilon_{c}$ changes, and $\eta_{c}$ does not change. In the graph
${\mathcal{G}}^{3}$, there is no sign change (we can assume $w,b$ are paired
in ${\mathcal{G}}^{3}$). In the graph ${\mathcal{G}}^{1}$ or
${\mathcal{G}}^{2}$, there is a sign change of $(-1)^{(\ell-1)/2}$ and
$(-1)^{(\ell+1)/2}$ respectively, resulting in $(-1)^{\ell}=-1$, a net sign
change. Consequently $\varepsilon_{c}\delta_{c}\eta_{c}$ is invariant in this
case as well. ∎

###  2.9 Partition function

Fix ${\mathcal{G}}$ as before with $n$ nodes, satisfying (3), $k$ of which are
white and $n-k$ of which are black. Let $\Omega$ be the set of multiwebs in
${\mathcal{G}}$.

We define the _partition function_ $Z=Z({\mathcal{G}})$ by

$Z=\sum_{m\in\Omega}\nu(m)\mathrm{Tr}_{m}=\sum_{m\in\Omega}\nu(m)\sum_{\lambda\in\Lambda}C_{\lambda}(m)\mathrm{Tr}_{\lambda}$
(8)

where the $C_{\lambda}(m)$ are defined in (2). Then $Z$ is itself an
$\text{SL}_{3}$-invariant multilinear function on $Y^{\otimes
k}\otimes(Y^{*})^{\otimes n-k}$. By interchanging the order of summation we
can write

$Z=\sum_{\lambda\in\Lambda}C_{\lambda}\mathrm{Tr}_{\lambda}$ (9)

where $C_{\lambda}=\sum_{m\in\Omega}\nu(m)C_{\lambda}(m).$ We call
$C_{\lambda}$ the _partition function for class $\lambda$_.

Our main result is a computation of $C_{\lambda}$. We show that

$C_{\lambda}=\Delta^{3}P_{\lambda}$ (10)

where $P_{\lambda}$ is a certain integer-coefficient polynomial in the entries
in the boundary measurement matrix $X$, and $\Delta$, which is independent of
$\lambda$, is the determinant of a submatrix of the associated Kasteleyn
matrix, as defined in Section 2.7 above.

If $c:V_{\partial}\to{\mathcal{C}}$ be a feasible coloring of the nodes, then
$Z(c)$ is a number, and

$Z(c)=\sum_{m\in\Omega}\nu(m)\mathrm{Tr}_{m}(c)=\sum_{\lambda\in\Lambda}C_{\lambda}\mathrm{Tr}_{\lambda}(c).$
(11)

By Lemma 1, for feasible $c$, all terms in (11) have the same sign, and so
$Z(c)$ defines a natural probability measure $\text{Pr}_{c}$ on $\Lambda$,
with

$\text{Pr}_{c}(\lambda)=\frac{C_{\lambda}\mathrm{Tr}_{\lambda}(c)}{Z(c)}.$
(12)

This is the probability that a “random reduction” of a random multiweb (with
boundary colors $c$) will have shape $\lambda$. More precisely, choose a
($\mathrm{Tr}_{c}$-weighted) random multiweb $m$ with boundary colors $c$, and
consider all possible reduced webs (with multiplicity) after applying skein
relations to $[m]$. Choose one of these uniformly at random, and consider its
shape $\lambda$.

We have an explicit determinantal expression for $Z(c)$. Let
$V_{\partial}=V_{1}\cup V_{2}\cup V_{3}$ be the partition of $V_{\partial}$
into the nodes of color $1,2,3$ respectively. Let ${\mathcal{G}}_{i}$ be the
graph obtained from ${\mathcal{G}}$ by removing nodes not in $V_{i}$. Note
that ${\mathcal{G}}_{i}$ is balanced: it has the same number of black vertices
as white vertices, by (4). Let $Z_{i}=|\det K({\mathcal{G}}_{i})|$, the
determinant of the Kasteleyn matrix of ${\mathcal{G}}_{i}$. Here we can take
$K({\mathcal{G}}_{i})$ to be the Kasteleyn matrix of ${\mathcal{G}}$
restricted to ${\mathcal{G}}_{i}$, since removing vertices from the outer face
of ${\mathcal{G}}$ yields a submatrix of $K$ also satisfying the Kasteleyn
condition.

######  Lemma 3.

We have $Z(c)=\varepsilon_{c}Z_{1}Z_{2}Z_{3}$ where $\varepsilon_{c}$ is the
sign from Lemma 1.

###### Proof.

By (11), $Z(c)$ is weighted sum of multiwebs, multiplied by their traces. By
Lemma 1, the trace is a global sign $\varepsilon_{c}$ times the number of
colorings. So $Z(c)$ is $\varepsilon_{c}$ times the weighted number of colored
multiwebs with node colors $c$. A colored multiweb with node colors $c$ is the
same as a $3$-tuple of dimer covers, one of each color, where the dimer cover
of color $i$ is a dimer cover of the graph ${\mathcal{G}}_{i}$. But
$Z_{i}=|\det K({\mathcal{G}}_{i})|$ is the positive weighted sum of dimer
covers of ${\mathcal{G}}_{i}$. ∎

By definition of $\delta_{c}$ we can write
$Z_{1}Z_{2}Z_{3}=\varepsilon_{K}\delta_{c}\det K({\mathcal{G}}_{1})\det
K({\mathcal{G}}_{2})\det K({\mathcal{G}}_{3})$ where $\varepsilon_{K}$ is the
sign of $\det K$. Thus

$Z(c)=\varepsilon_{c}\delta_{c}\varepsilon_{K}\det K({\mathcal{G}}_{1})\det
K({\mathcal{G}}_{2})\det K({\mathcal{G}}_{3}).$ (13)

##  3 Examples

We illustrate here the computation of partition functions and probabilities
for reduced webs in the cases when ${\mathcal{G}}$ has $4$ and $5$ nodes.
Cases with $2$ or $3$ nodes are trivial, since there is only one reduced web
class in these cases, see Figure 12. In the appendix we illustrate the cases
with $6$ nodes.

###  3.1 $(w,b,w,b)$

When $n=4$, by (3) we must have $|W_{\partial}|=|B_{\partial}|=2$, and there
are two possible types, up to rotational symmetry, which are $p=(w,b,w,b)$ and
$p=(w,w,b,b)$.

Suppose $p=(w,b,w,b)$. The matrix $X$ is $2\times 2$. The set $\Lambda$
consists of two classes of reduced webs, $\lambda_{1},\lambda_{2}$, shown in
Figure 13.

Figure 13: The two classes of webs with $4$ boundary points and $p=(w,b,w,b)$.

As a consequence of Theorem 4 below, we have (up to a global sign)

$P_{\lambda_{1}}=X_{11|22}=X_{1,1}X_{2,2}$ (14)

and

$P_{\lambda_{2}}=X_{12|21}=-X_{1,2}X_{2,1}$ (15)

where $P_{\lambda}$ is defined in (10).

For example take color $c\equiv 1$ for the four nodes. The traces of both webs
are $1$. The probabilities of the two webs are then

$Pr_{c}(\lambda_{1})=\frac{X_{1,1}X_{2,2}}{X_{1,1}X_{2,2}-X_{1,2}X_{2,1}},~{}~{}~{}Pr_{c}(\lambda_{2})=\frac{-X_{1,2}X_{2,1}}{X_{1,1}X_{2,2}-X_{1,2}X_{2,1}}.$
(16)

For a concrete setting for this example, consider the graph ${\mathcal{G}}$ of
Figure 14, with edge weights indicated.

Figure 14: Graph with four nodes and $p=(w,b,w,b)$.

Then $\Omega_{c}({\mathcal{G}})$ has $3$ multiwebs, two of which are already
reduced and the third of which is reducible into one of each web class. We
have

$Z(c)=abde^{2}g^{2}k+abdf^{2}h^{2}k+2abdefghk$ (17)

which is a sum of the weights (times traces) of the three multiwebs. The last
term corresponds to a non-reduced web with a quad face, which can be reduced
to one copy of each reduced web class. We thus have

$\displaystyle C_{\lambda_{1}}$ $\displaystyle=abde^{2}g^{2}k+abdefghk$
$\displaystyle C_{\lambda_{2}}$ $\displaystyle=abdf^{2}h^{2}k+abdefghk.$

A Kasteleyn matrix for the given vertex order is

$K=\begin{pmatrix}0&0&b&0\\\ 0&0&0&d\\\ a&0&e&h\\\ 0&k&-f&g\end{pmatrix}$

for which we find

$Z(c)=Z({\mathcal{G}})Z({\mathcal{G}}_{int})^{2}=abdk(eg+fh)^{2},$

equal to (17), and the corresponding reduced Kasteleyn matrix is

$X=\frac{1}{\Delta}\begin{pmatrix}-abg&bkh\\\ -adf&-kde\end{pmatrix}$

where $\Delta=\det D=\det\begin{pmatrix}e&f\\\ -h&g\end{pmatrix}=eg+fh.$ From
this one can verify (14) and (15). Note that a different choice of Kasteleyn
signs might change the sign of (both of) $P_{\lambda_{1}},P_{\lambda_{2}}$ but
the probabilities remain unchanged.

Suppose we had chosen a different set of node colors $c^{\prime}$, for example
color $1$ at black and white nodes $1$ and color $2$ at black and white nodes
$2$. Then $\lambda_{2}$ is incompatible with $c^{\prime}$, and so
$\mathrm{Tr}_{\lambda_{2}}(c)=0$. We still have
$\mathrm{Tr}_{\lambda_{1}}(c)=1$. In this case

$\displaystyle Z_{c^{\prime}}$
$\displaystyle=Z({\mathcal{G}}\setminus\\{2,{\bf
2}\\})Z({\mathcal{G}}\setminus\\{1,{\bf 1}\\})Z({\mathcal{G}}_{int})$
$\displaystyle=(abg)(dek)(eg+fh)$ $\displaystyle=abde^{2}g^{2}k+abdefghk$
$\displaystyle=C_{\lambda_{1}}$
$\displaystyle=C_{\lambda_{1}}\mathrm{Tr}_{\lambda_{1}}(c^{\prime})+C_{\lambda_{2}}\mathrm{Tr}_{\lambda_{2}}(c^{\prime}).$

###  3.2 $(w,w,b,b)$

The other case with $n=4$ has $p=(w,w,b,b)$. Again $\Lambda$ has two different
reduced web classes, shown in Figure 15.

Figure 15: The two classes of webs with $4$ boundary points and $p=(w,w,b,b)$.

Again $X$ is a $2\times 2$ matrix. Theorem 4 shows that, up to a global sign,

$P_{\lambda_{1}}=X_{1,2}X_{2,1}$

and

$P_{\lambda_{2}}=X_{1,1}X_{2,2}-X_{1,2}X_{2,1}.$

As a concrete realization, let ${\mathcal{G}}$ be the $3\times 2$ grid graph
of Figure 16. A Kasteleyn matrix is

$K=\begin{pmatrix}a&0&b\\\ 0&k&d\\\ -g&-e&f\end{pmatrix}$

and

$X=\begin{pmatrix}a+\frac{bg}{f}&\frac{be}{f}\\\
\frac{dg}{f}&k+\frac{de}{f}\end{pmatrix}$

with $\Delta=f$.

Figure 16: Graph with four nodes and $p=(w,w,b,b)$.

###  3.3 $(b,w,w,w,w)$

When $n=5$ there is, up to rotation, only one possible type pattern,
$p=(b,w,w,w,w)$. In this case $\Lambda$ consists in 3 classes
$\lambda_{1},\lambda_{2},\lambda_{3}$, shown in Figure 17. Now $X$ is a
$4\times 2$ matrix, and Theorem 4 shows that, up to a global sign,

$\displaystyle P_{\lambda_{1}}$
$\displaystyle=X_{1,2}X_{2,2}(X_{3,1}X_{4,2}-X_{4,1}X_{3,2})$ $\displaystyle
P_{\lambda_{2}}$ $\displaystyle=X_{3,2}X_{4,2}(X_{1,1}X_{2,2}-X_{2,1}X_{1,2})$
(18) $\displaystyle P_{\lambda_{3}}$
$\displaystyle=X_{1,2}X_{4,2}(X_{2,1}X_{3,2}-X_{3,1}X_{2,2}).$

Figure 17: Classes of webs with $5$ boundary points and $p=(b,w,w,w,w)$.

##  4 Reduction matrix

###  4.1 The reduction matrix ${\cal P}$

For a fixed type vector $p$, we define a matrix ${\cal P}={\cal P}(p)$ with
rows indexed by reduced web classes $\Lambda$ and columns indexed by
$3$-partitions $\Pi$, as follows. Using the basic skein relation (7) each
$3$-partition $\tau\in\Pi$ can be converted into a unique formal linear
combination of planar web classes $\lambda\in\Lambda$:

$\tau=\sum_{\lambda\in\Lambda}{\cal P}_{\lambda,\tau}\lambda.$ (19)

That is, each $3$-partition can be represented as a nonplanar diagram in a
disk, as in Section 2.6. Now replace each crossing with the corresponding
linear combination of the two locally planar resolutions. Reduce each
resulting planar diagram using the planar skein relations (6). The resulting
linear combination of reduced planar webs is unique, independent of the
original drawing of the $3$-partition in the disk (only depending on its
isotopy class) and independent of the order of reductions. We call the matrix
${\cal P}$ the _reduction matrix from $3$-partitions to reduced web classes_,
or _reduction matrix_ for short.

As an example, consider the case $n=5$ with $p=(b,w,w,w,w)$. Ordering
$\Lambda$ as in Figure 17, and using $\Pi=\\{{\bf 1}1|234,{\bf 1}2|134,{\bf
1}3|124,{\bf 1}4|123\\}$ (where boldface denotes black nodes), the reduction
matrix ${\cal P}$ is given in table 1.

${\bf 1}1|234$ ${\bf 1}2|134$ ${\bf 1}3|124$ ${\bf 1}4|123$ $1$ 0 0 1 1 $2$ 1
1 0 0 $3$ 0 -1 -1 0

Table 1:  The reduction matrix for $p=(b,w,w,w,w)$.

###  4.2 $X_{\tau}$ variables

For $\tau\in\Pi$ define variables $X_{\tau}$ as follows. First, if there are
no triples, that is, if $|W_{\partial}|=|B_{\partial}|$, then we can think of
$\tau$ as a bijection $\sigma$ from white nodes to black nodes, that is, from
$[n]$ to $[n]$ (identifying nodes with rows and columns of $X$) and in this
case

$X_{\tau}=(-1)^{\sigma}\prod_{w\in W_{\partial}}X_{w\sigma(w)}.$ (20)

This is consistent with the definition in [KW11].

If there are triples in $\tau$, the matrix $X$ is not square. In $X$ replace
each column for $b\in B_{int}^{*}$ with three consecutive identical columns.
This yields a square matrix $\tilde{X}$. For each $b\in B_{int}^{*}$, we now
treat $b$ as three nodes which are three copies of itself: we add three
consecutive black nodes $b_{1},b_{2},b_{3}$ to the set $B_{\partial}$ (in
clockwise order, at some point along the boundary).

Let $\Sigma$ be the set of bijections from triples of $\tau$ to $B_{int}^{*}$.
Given $\tau$ and a bijection $\alpha\in\Sigma$ we extend $\alpha$ to a
bijection (pairing) $\pi$ from the set of all white nodes to the augmented set
of black nodes, by sending, for each triple $t=w_{1}w_{2}w_{3}$, the nodes
$w_{1},w_{2},w_{3}$ to $b_{1},b_{2},b_{3}$ in that order, where $b=\alpha(t)$
(the pairs of $\tau$ are still paired in $\pi$). See Figure 18.

Figure 18: Converting triples to pairings. On the left, a triple $t$ of
$\tau$, with center vertex $b=\alpha(t)$. On the right, the corresponding
pairing: we push $b$ to the boundary, replacing it with three new consecutive
nodes $b_{1},b_{2},b_{3}$ in clockwise order, connecting the $w_{i}$s to the
$b_{i}$s.

Define

$X_{\tau}:=\sum_{\alpha\in\Sigma}\tilde{X}_{\pi}$ (21)

where $\pi=\pi(\alpha)$, and $\tilde{X}_{\pi}$ is defined for the pairing
$\pi$ as in (20). Note that if we change $\pi$ to $\pi^{\prime}$ by changing
the bijection between the $w_{1},w_{2},w_{3}$ and the $b_{1},b_{2},b_{3}$,
then $\tilde{X}_{\pi^{\prime}}=\pm\tilde{X}_{\pi}$ changes only by the sign of
this bijection, since the corresponding columns $b_{1},b_{2},b_{3}$ of
$\tilde{X}$ are identical.

In the above example, where $p=(b,w,w,w,w)$, the matrix $X$ is $4\times 2$. We
have

$\displaystyle X_{{\bf 1}1|234}$ $\displaystyle=X_{11}X_{22}X_{32}X_{42}$
$\displaystyle X_{{\bf 1}2|134}$ $\displaystyle=-X_{21}X_{12}X_{32}X_{42}$
(22) $\displaystyle X_{{\bf 1}3|124}$ $\displaystyle=X_{31}X_{12}X_{22}X_{42}$
$\displaystyle X_{{\bf 1}4|123}$ $\displaystyle=-X_{41}X_{12}X_{22}X_{32}.$

Note that if we relabel the second indices $2$ with $2,3,4$ (to correspond to
their column in $\tilde{X}$) the sign is $(-1)^{\pi}$.

###  4.3 Main result

######  Theorem 4.

We have $P_{\lambda}=\varepsilon_{K}\sum_{\tau\in\Pi}{\cal
P}_{\lambda,\tau}X_{\tau},$ where $\varepsilon_{K}$ is a sign independent of
$\lambda$, depending only on the choice of Kasteleyn signs.

As an example, combining table (1) with (22) we get (18): for example

$P_{\lambda_{1}}=X_{{\bf 1}3|124}+X_{{\bf
1}4|123}=X_{31}X_{12}X_{22}X_{42}-X_{41}X_{12}X_{22}X_{32}.$

###### Proof of Theorem 4.

Suppose first that $|W_{\partial}|=|B_{\partial}|$, so there are no triples.
Let $c$ have $p,q$ and $r$ white nodes of color $1,2,3$ respectively (and thus
$p,q,r$ black nodes of color $1,2,3$ also). Let $W_{i}\subset W_{\partial}$ be
the white nodes of color $i$ and $B_{i}\subset B_{\partial}$ be the black
nodes of color $i$. By Lemma 3, the partition function is
$Z(c)=\varepsilon_{c}Z_{1}Z_{2}Z_{3}$.

We write (see Lemma 3 and (13))

$\displaystyle Z(c)$ $\displaystyle=\varepsilon_{c}Z_{1}Z_{2}Z_{3}$
$\displaystyle=\varepsilon_{c}\varepsilon_{K}\delta_{c}\det
K({\mathcal{G}}_{1})\det K({\mathcal{G}}_{2})\det K({\mathcal{G}}_{3}).$

Thus with $X_{i}=X_{W_{i}}^{B_{i}}$ we have

$\displaystyle\frac{Z(c)}{\Delta^{3}}$
$\displaystyle=\varepsilon_{c}\varepsilon_{K}\delta_{c}\det X_{1}\det
X_{2}\det X_{3}$
$\displaystyle=\varepsilon_{c}\varepsilon_{K}\delta_{c}\Big{(}\sum_{\sigma_{1}}(-1)^{\sigma_{1}}X_{i_{1}\sigma_{1}(i_{1})}\dots
X_{i_{p}\sigma_{1}(i_{p})}\Big{)}\Big{(}\sum_{\sigma_{2}}(-1)^{\sigma_{2}}(\dots)\Big{)}\Big{(}\sum_{\sigma_{3}}(-1)^{\sigma_{3}}(\dots)\Big{)}$
$\displaystyle=\varepsilon_{c}\varepsilon_{K}\delta_{c}\eta_{c}\sum_{\tau\in\Pi}(-1)^{\tau}X_{1\tau_{1}}\dots
X_{n\tau_{n}}\mathrm{Tr}_{\tau}(c)$
$\displaystyle=\varepsilon_{K}\sum_{\tau\in\Pi}X_{\tau}\mathrm{Tr}_{\tau}(c)$
(23)

where in the third line we used the definition of $\eta_{c}$ and in the last
line we used Lemma 2.

Now (23), which holds for each $c$, writes the function $Z$ as a linear
combination of traces $\mathrm{Tr}_{\tau}$. Using (19), we have

$\mathrm{Tr}_{\tau}=\sum_{\lambda}{\cal
P}_{\lambda,\tau}\mathrm{Tr}_{\lambda}.$

Plugging this into (23),

$\frac{Z}{\Delta^{3}}=\varepsilon_{K}\sum_{\tau\in\Pi}X_{\tau}\sum_{\lambda\in\Lambda}{\cal
P}_{\tau,\lambda}\mathrm{Tr}_{\lambda}.$

Interchanging the order of summation,

$\frac{Z}{\Delta^{3}}=\varepsilon_{K}\sum_{\lambda\in\Lambda}(\sum_{\tau\in\Pi}X_{\tau}{\cal
P}_{\tau,\lambda})\mathrm{Tr}_{\lambda}.$

Since the $\\{\mathrm{Tr}_{\lambda}\\}_{\lambda\in\Lambda}$ are independent
functions (see Section 2.5 and [Kup96]), this allows us to identify
$P_{\lambda}$ with the coefficient of $\mathrm{Tr}_{\lambda}$ in this sum,
that is, $P_{\lambda}=\varepsilon_{K}\sum_{\tau\in\Pi}X_{\tau}{\cal
P}_{\tau,\lambda}$ as desired.

Now suppose we have $|W_{\partial}|>|B_{\partial}|$. We reduce this case to
the previous case, as follows. Recall the definition of $W^{*}$ from Section
2.1; see Figure 4. We add a spike to each $w\in W^{*}$, splitting $w$ into two
white vertices $w,w_{s}$ connected by a black vertex $b$ of degree $2$, so
that neighbors of $w$ are now neighbors of $w_{s}$, and $w$ is adjacent only
to $b$. The new edges have weight $1$. See Figure 19, left. We can then assume
that $B_{int}^{*}$ consists of these new $b$ vertices.

We now add a “gadget” to ${\mathcal{G}}$ consisting of, for each $w\in W^{*}$,
three new consecutive black nodes $b_{1},b_{2},b_{3}$, and one white internal
vertex, located on the same side of $w$ as the vertex $w^{\prime}\in W^{**}$
associated to $w$. These are connected as shown in Figure 19. The edges
connecting $b_{1},b_{3}$ to $w$ and $w^{\prime\prime}$ have weight
$\varepsilon$; other new edges have weight $1$.

Figure 19: We add spikes at $w_{2},w_{4}$ splitting them into segments
$w_{2},w_{s2}$ and $w_{4},w_{s4}$. We then add gadgets to
$B_{int}^{*}=\\{b_{4},b_{5}\\}$ as shown. Dotted edges are the edges of weight
$\varepsilon$.

This augmented graph $\tilde{\mathcal{G}}$ has a dimer cover (extending the
one of the figure to each gadget in a unique way). Moreover in the limit
$\varepsilon\to 0$ the $X$ matrix $\tilde{X}$ for $\tilde{\mathcal{G}}$, which
is square, is related to the $X$ matrix of ${\mathcal{G}}$: For each $b\in
B_{int}^{*}$ we replace the corresponding column of $X$ with three consecutive
columns, labelled $b_{1},b_{2},b_{3}$, each equal (in the limit
$\varepsilon\to 0$) to the initial column, to get $\tilde{X}$.

We can now proceed as in the case $|W_{\partial}|=|B_{\partial}|$ above. Let
$c$ be a coloring of the nodes of ${\mathcal{G}}$. We extend this to a
coloring $\tilde{c}$ of $\tilde{\mathcal{G}}$ by coloring each triple
$b_{1},b_{2},b_{3}$ by colors $1,2,3$ in that order. From (23),

$\varepsilon_{K}\frac{Z(\tilde{c})}{\Delta^{3}}=\sum_{\rho\in\tilde{\Pi}}\tilde{X}_{\rho}\mathrm{Tr}_{\rho}(\tilde{c}).$
(24)

Here $\tilde{\Pi}$ consists of node pairings of the augmented set of nodes. We
group this sum into terms arising from each $\tau\in\Pi$: for each
$\tau\in\Pi$ and bijection $\alpha\in\Sigma$ there is unique associated
pairing $\rho\in\tilde{\Pi}$ with nonzero trace (since for each triple the
white node of color $1$ must go to $b_{1}$, the white node of color $2$ must
go to $b_{2}$, and the white node of color $3$ must go to $b_{3}$). The sum in
(24) is

$=\sum_{\tau\in\Pi}\sum_{\alpha\in\Sigma}\tilde{X}_{\rho}\mathrm{Tr}_{\rho}(\tilde{c})$

where $\rho=\rho(\tau,\alpha)$ is the associated pairing.

Recall that $X_{\tau}=\sum_{\alpha}\tilde{X}_{\pi}$ (see (21) where $\pi$
pairs each triple $w_{1},w_{2},w_{3}$ with $b_{1},b_{2},b_{3}$ in the natural
order. Let us compare $\tilde{X}_{\rho}$ with $\tilde{X}_{\pi}$: in
$\tilde{X}_{\rho}$, the $w_{1},w_{2},w_{3}$ are paired with the
$b_{1},b_{2},b_{3}$ so as to match the colors $\tilde{c}$; the node among
$w_{1},w_{2},w_{3}$ of color $1$ is paired with $b_{1}$ and so on. Thus
$\tilde{X}_{\rho}=\pm\tilde{X}_{\pi}$ where the sign is $+$ if and only if the
permutation from $w_{1},w_{2},w_{3}$ to $b_{1},b_{2},b_{3}$ is even, that is,
if nodes $w_{1},w_{2},w_{3}$ are colored in counterclockwise order. Note also
that this is the sign of the contribution of this triple to
$\mathrm{Tr}_{\tau}(c)$. Thus

$\sum_{\alpha}\tilde{X}_{\rho}\mathrm{Tr}_{\rho}(\tilde{c})=\sum_{\alpha}X_{\pi}\mathrm{Tr}_{\tau}(c)=X_{\tau}\mathrm{Tr}_{\tau}(c).$

So the sum is

$\varepsilon_{K}\frac{Z(c)}{\Delta^{3}}=\varepsilon_{K}\frac{Z(\tilde{c})}{\Delta^{3}}=\sum_{\tau\in\Pi}\sum_{\alpha\in\Sigma}\tilde{X}_{\rho}\mathrm{Tr}_{\rho}(\tilde{c})=\sum_{\tau\in\Pi}X_{\tau}\mathrm{Tr}_{\tau}(c).$

∎

##  5 Web pairing matrices

The reduction matrix ${\cal P}$ is related to two other matrices:

${\mathcal{M}}{\cal P}={\mathcal{E}}$

where ${\mathcal{M}}$ is the “web pairing matrix” and ${\mathcal{E}}$ is the
“extended web pairing matrix”, both of which we now define. Rows and columns
of ${\mathcal{M}}$ are indexed by reduced webs and
${\mathcal{M}}_{r_{1},r_{2}}$ is the trace of the web drawn on the sphere with
$r_{1}$ inside the disk and $r_{2}$ outside the disk (reflected in the
bounding circle from the inside to the outside the disk, so that the nodes of
$r_{2}$ correspond to the nodes of $r_{1}$ in the same cyclic order). In the
$(w,b,w,b,w,b)$ example of Section 9.1 we have

${\mathcal{M}}=\begin{pmatrix}27&9&9&3&9&24\\\ 9&27&3&9&3&24\\\
9&3&27&9&3&24\\\ 3&9&9&27&9&24\\\ 9&3&3&9&27&24\\\
24&24&24&24&24&72\end{pmatrix}.$

Rows of ${\mathcal{E}}$ are indexed by reduced webs and columns are indexed by
$3$-partitions $\Pi$. The entry ${\mathcal{E}}_{r,\pi}$ is the trace of the
web drawn on the sphere with $r$ inside the disk and $\pi$ outside the disk,
reflected as above. Since $\pi$ is not necessarily planar, the trace
${\mathcal{E}}_{r,\pi}$ may be negative or zero. In the
$(w,b,w,b,w,b)$-example we have

${\mathcal{E}}=\begin{pmatrix}27&9&9&3&9&3\\\ 9&27&3&9&3&9\\\ 9&3&27&9&3&9\\\
3&9&9&27&9&3\\\ 9&3&3&9&27&9\\\ 24&24&24&24&24&0\end{pmatrix}.$

######  Theorem 5.

Let ${\mathcal{M}}$ be the web trace matrix and ${\mathcal{E}}$ the extended
web trace matrix. Then ${\mathcal{M}}{\cal P}={\mathcal{E}}$.

###### Proof.

Let $\lambda\in\Lambda$ and $\tau\in\Pi$. By definition
${\mathcal{E}}_{\lambda,\tau}$ is the trace of a graph drawn on the sphere,
which is $\lambda$ in the lower half sphere and $\tau$ in the upper half
sphere, reflected as mentioned above. We use skein relations in the upper half
sphere to replace $\tau$ with a linear combination of planar webs,
$\tau=\sum_{\lambda^{\prime}\in\Lambda}{\cal
P}_{\lambda^{\prime},\tau}r^{\prime}$, and then apply the trace after
extending by $\lambda$ in the lower half sphere, getting

${\mathcal{E}}_{\lambda,\tau}=\sum_{\lambda^{\prime}\in\Lambda}{\mathcal{M}}_{\lambda,\lambda^{\prime}}{\cal
P}_{\lambda^{\prime},\tau}$

as desired. ∎

We conjecture that ${\mathcal{M}}$ is invertible, in which case ${\cal
P}={\mathcal{M}}^{-1}{\mathcal{E}}$. The matrices
${\mathcal{M}},{\mathcal{E}}$ are the $\text{SL}_{3}$ analogs of the $q=2$
“meander matrix” $M$ and “extended meander matrix $E$, which are relevant for
$\text{SL}_{2}$ connections, see [KW11]. The meander matrix $M$ is indexed
with planar pairings rather than planar reduced webs, but otherwise has a
similar definition: $M_{\pi,\pi^{\prime}}$ is the trace of the $2$-web on the
sphere formed from $\pi$ in the upper hemisphere and $\pi^{\prime}$ in the
lower hemisphere. Likewise for $E$. The trace in this case is just $2^{c}$
where $c$ is the number of components of the $2$-web. It is shown in [KS91]
that $M$ is invertible. In [DF97], di Francesco discusses a generalization of
the meander matrix, called the $\text{SU}(n)$ meander matrix, and shows it is
invertible for each $n$. We don’t know if it is related to our matrix.

##  6 Parallel crossings

The Lindstrøm-Gessel-Viennot Theorem (see e.g. [Sta99]) is a determinantal
formula for the number of pairwise disjoint monotone lattice paths in a planar
lattice (with appropriate boundary conditions). It has a number of
generalizations, notably by Fomin [Fom01] to the case of spanning forests (or
loop-erased walks), and by Kenyon and Wilson [KW11] to the case of double
dimer paths.

We give a generalization here for reduced webs consisting of parallel
crossings. Remarkably our formula involves a product of two determinants,
rather than a single determinant.

Let ${\mathcal{G}}$ be a circular planar graph, with $2n$ nodes
$v_{1},\dots,v_{2n}$. Suppose for each $i$ that $v_{i}$ and $v_{2n+1-i}$ have
opposite type: one is white and the other black. Consider the reduced web
$\lambda_{\tiny{\|}}$ which pairs $i$ to $2n+1-i$, that is, represents the
“parallel” crossing of a rectangle (see Figure 20).

Figure 20: The “parallel crossing” reduced web.

Let $W_{1},W_{2}$ be the set of indices of white nodes on the left and right,
respectively, and likewise $B_{1},B_{2}$ the set of black nodes on the left
and right. Let $X_{W_{1}}^{B_{2}}$ be the submatrix of $X$ with rows indexing
$W_{1}$ and columns indexing $B_{2}$.

######  Theorem 6.

We have $P_{\lambda_{\tiny{\|}}}=\det X_{W_{1}}^{B_{2}}\det
X_{W_{2}}^{B_{1}}.$

For example, see Figure 15, second panel, for which (after a rotation)
$W_{1}=\\{1,2\\}$ and $B_{1}=\emptyset$ and whose polynomial is $\det X$. See
also the third configuration of Figure 30, whose polynomial is
$X_{2,3}\begin{vmatrix}X_{1,1}&X_{1,2}\\\ X_{3,1}&X_{3,2}\end{vmatrix}$ (we
changed indices to match those of the figure.)

If $c\equiv 1$, the probability of the parallel crossing, for the natural
measure on reduced webs, is $Pr(\lambda_{\tiny{\|}})=\frac{\det
X_{W_{1}}^{B_{2}}\det X_{W_{2}}^{B_{1}}}{\det X}.$

###### Proof of Theorem 6.

By Theorem 4 we need to compute ${\cal P}_{\lambda,\tau}$ for
$\lambda=\lambda_{\tiny{\|}}$ and any pairing $\tau$. For clarity draw
${\mathcal{G}}$ in a rectangle so that nodes $1,\dots,n$ are on the left side
and nodes $n+1,\dots,2n$ are on the right side.

Let $\tau\in\Pi$ be a pairing of nodes. From the definition of ${\cal P}$, we
need to reduce $\tau$ to a linear combination of planar pairings using skein
relations and see when the parallel crossing $\lambda$ occurs in this
reduction. First suppose that $\tau$ pairs two nodes $w,b$ which are both on
the left side of the rectangle. By induction on the distance from $w$ to $b$
we will show that no reduction of $\tau$ contains $\lambda$, and thus ${\cal
P}_{\lambda,\tau}=0$. Assume that no two nodes on the left side between $w$
and $b$ are paired: there is no nested pairing inside $wb$. If $w,b$ are
adjacent, they never participate in any skein relation, so $w,b$ are paired in
any planar reduction of $\tau$, so ${\cal P}_{\lambda,\tau}=0$. If $w,b$ are
not adjacent, we “comb” the web $\tau$, isotoping it so that there are no
crossings of pairs in $\tau$ between the strand $wb$ and the left boundary, as
shown in Figure 21. That is, move all crossings enclosed between the strand
$wb$ and the left boundary across the strand $wb$.

Figure 21: Draw the web so that there are no crossings of strands between the
left boundary and the strand $wb$.

Suppose $w$ lies above $b$ on the left boundary. If the first node below $w$
is also white, say $w^{\prime}$, then applying a skein relation to the local
crossing between $w$ and $w^{\prime}$ results in two webs; the first connects
$w^{\prime}$ to $b$ and can be eliminated by induction. The second has a
double-Y connecting $w,w^{\prime}$ to an inner black vertex $b^{\prime}$.
These three vertices persist under any further skein reductions, showing that
${\cal P}_{\lambda,\tau}=0$. We can therefore assume the first node below $w$
is black, say $b_{1}$. There are now three cases: the next node below $b_{1}$
is $b$, or another black node $b_{2}$, or a white node $w_{1}$. If the next
node below $b_{1}$ is $b$, then apply the skein relation to the intersection
of the strand $wb$ with the strand from $b_{1}$; neither resulting web
contributes to ${\cal P}$ as above. Suppose the next node below $b_{1}$ is a
black node $b_{2}\neq b$; the white case is similar. Let $x_{1},x_{2}$ denote
the crossings between $wb$ and the strand starting at $b_{1}$, and between
$wb$ and the strand starting at $b_{2}$. Apply a skein relation to the
crossing $x_{1}$. Replacing $x_{1}$ with a parallel crossing produces an
adjacent pair which persists under all further reductions as above. So we must
replace the crossing $x_{1}$ with a double-Y. Now apply the skein relation to
the crossing $x_{2}$. Replacing it with a parallel crossing results in
$b_{1},b_{2}$ being connected to an internal vertex $w^{\prime}$ and these
three persist for all future reductions. So we must replace $x_{2}$ with a
double-Y to get a nonzero contribution. The edge between these two double-Ys
(the thick blue edge in Figure 22) will persist under all further reductions
unless it is, at some point, part of a quad move (Figure 6, last line)
involving the face immediately to its right.

Figure 22: Resolving the crossings at $x_{1},x_{2}$.

However the two possible resolutions of this quad result, in one case in an
adjacent pair $wb_{1}$, and in the other case a pair of adjacent black nodes
$b_{1}b_{2}$ connected to an internal white vertex; both of these cases we can
eliminate as above. This completes the proof that ${\cal P}_{\lambda,\tau}=0$
when $\tau$ pairs two nodes $w,b$ both on the same side.

Now suppose that $\tau$ only pairs nodes from opposite sides. We claim that
${\cal P}_{\lambda,\tau}=1$. If $\tau$ is planar (has no crossings), it must
be $\lambda$. Consider the pairs in $\tau$ with white nodes on the left; if
none of these cross each other, and the same is true for the pairs with black
nodes on the left, then $\tau$ is planar. So if $\tau$ is nonplanar it must
have two pairs which cross, and whose left nodes have the same type (either
both black or both white), say white. In this case we can use the skein
relation to replace this crossing with a parallel uncrossing, and another web
which connects two white vertices $w_{1},w_{2}$ on the left to a black vertex
$b$ in the interior. We claim by induction on the distance between
$w_{1},w_{2}$ that such a web contributes zero. If $w_{1},w_{2}$ are adjacent,
see as above. If not, comb so that there are no crossings of other strands in
the region between $w_{1},b,w_{2}$ and the left boundary. Further assume that
strands for all nodes between $w_{1},w_{2}$ exit this region between $w_{1}$
and $b$, that is, cross $w_{1}b$. Then we can argue exactly as in the previous
case of a pairing between a black and white vertex on the left by making the
strand $w_{2}b$ very small $b$ acts as if it is on the left boundary.

Consequently every crossing between pairs with the same type on the left must
resolve to a _parallel uncrossing_ , in order to contribute nontrivially to
$P_{\lambda,\tau}$. Resolving all such pairs results in the web $\lambda$,
with a sign $+$. Therefore ${\cal P}_{\lambda,\tau}=1$ for all pairings
connecting nodes on opposite sides. Therefore $\sum_{\tau}{\cal
P}_{\lambda,\tau}X_{\tau}=\sum_{\tau\in\Pi^{\prime}}X_{\tau}$ where
$\Pi^{\prime}$ is the set of $\tau$’s pairing nodes on opposite sides. Using
the definition of $X_{\tau}$ from (20) this completes the proof. ∎

The above proof generalizes (thanks to Pasha Pylyavskyy for suggesting this)
to crossings with “crossbars”, as illustrated in Figure 23: take $n$ parallel
strands as in Figure 20, and draw any nonnegative number of crossbars between
adjacent strands with the proviso that the crossbars be _interlaced_ : between
two crossbars connecting strands $i$ and $i+1$ there is exactly one crossbar
from $i-1$ to $i$ and one from $i+1$ to $i+2$. Note that this produces a
reduced web, and the vertex types are determined on each connected component
up to a global type flip.

Figure 23: A crossing with crossbars (a crossbar is a vertical segment joining
two horizontal lines).

As above let $W_{1},W_{2}$ be the white nodes on the left and right and
$B_{1},B_{2}$ the black nodes on the left and right.

######  Theorem 7.

For a reduced web $\lambda$ which is a crossing with crossbars as defined
above, $P_{\lambda}=\det X_{W_{1}}^{B_{2}}\det X_{W_{2}}^{B_{1}}.$

###### Proof.

The first part of the proof, showing that $\tau$ cannot pair two nodes on the
same side, is identical to the first part of the proof of Theorem 6 above. So
we may suppose that $\tau$ only pairs nodes on opposite sides. In this case we
claim that ${\cal P}_{\lambda,\tau}=1$.

Consider a pair of crossing strands of $\tau$ with black nodes on the left and
white on the right, or vice versa. Applying the skein relation, turning a
crossing into a double-Y and a parallel uncrossing, the double-Y cannot
contribute to ${\cal P}_{\lambda,\tau}$ as discussed in the previous proof. So
we must resolve this crossing into two parallel strands (note that this does
not introduce any sign change). We resolve all crossings between strands with
the both left nodes black or both left nodes white.

We can now suppose that all strands of $\tau$ with white nodes on the left are
disjoint, and all strands of $\tau$ with black nodes on the left are disjoint.
Note that there is a unique such pairing $\tau$. Each crossing in $\tau$
involves two strands with different types on the left. The resolution of this
crossing must give a double-Y, since otherwise there would be a strand
connecting two vertices on the left. Resolving all crossings this way gives
$\lambda$ exactly.

In conclusion the pairings $\tau$ that contribute to ${\cal P}_{\lambda,\tau}$
are exactly those that pair nodes on opposite sides, and the contribution is
always $+1$. In particular $\sum_{\tau}{\cal P}_{\lambda,\tau}X_{\tau}$ is the
form in the statement. ∎

##  7 Application: triangular honeycombs

As another application of Theorem 4, we can compute the $P$-polynomial (and
probability) for the order-$n$ triangular honeycomb web $T_{n}$ of Figure 24
and the order-$n$ honeycomb $T^{\prime}_{n}$ of Figure 27. In the case of
$T_{n}$ and Figure 24 we used a different convention for the black nodes:
ordering them counterclockwise. This makes for a slightly less messy indexing
in the statement of Theorem 8.

###  7.1 The triangular honeycomb $T_{n}$

######  Theorem 8.

We have

$P_{T_{n}}=\det(\\{X_{i,j+n}\\}_{i,j=1..n})\det(\\{X_{i+n,j+2n}\\}_{i,j=1..n})\det(\\{X_{i+2n,j}\\}_{i,j=1..n}).$

For example for the order-$3$ example this is

$P_{T_{3}}=\det\begin{vmatrix}X_{1,4}&X_{1,5}&X_{1,6}\\\
X_{2,4}&X_{2,5}&X_{2,6}\\\ X_{3,4}&X_{3,5}&X_{3,6}\\\
\end{vmatrix}\det\begin{vmatrix}X_{4,7}&X_{4,8}&X_{4,9}\\\
X_{5,7}&X_{5,8}&X_{5,9}\\\
X_{6,7}&X_{6,8}&X_{6,9}\end{vmatrix}\det\begin{vmatrix}X_{7,1}&X_{7,2}&X_{7,3}\\\
X_{8,1}&X_{8,2}&X_{8,3}\\\ X_{9,1}&X_{9,2}&X_{9,3}\end{vmatrix}.$

Figure 24: Triangular honeycomb web $T_{n}$ of order $3$.

###### Proof of Theorem 8.

Let $\lambda$ denote the reduced web $T_{n}$. Let $I_{1},I_{2},I_{3}$ be the
three sides of $T_{n}$, with $I_{1}$ being the lower side as in the Figure:
$I_{1}$ consists of white and black nodes $\\{1,\dots,n\\}$, $I_{2}$ consists
of white and black nodes $\\{n+1,\dots,2n\\}$ and $I_{3}$ the remaining nodes.

The proof is similar to the proof of Theorem 6 above. Consider a pairing
$\tau\in\Pi$. In order to have ${\cal P}_{\lambda,\tau}\neq 0$, we claim that
$\tau$ must pair all white nodes in $I_{i}$ to black nodes in $I_{i+1}$, with
cyclic indices. Suppose not; suppose first that $\tau$ pairs some white node
$w\in I_{1}$ with a black node $b$ on the same side: $b\in I_{1}$. In this
case an induction argument identical to that in paragraphs 2 and 3 of the
proof of Theorem 6 implies that ${\cal P}_{\lambda,\tau}=0$. Secondly suppose
$\tau$ pairs a white node $w\in I_{1}$ with a black node $b\in I_{3}$. Suppose
$b$ is not the lower black vertex of $I_{3}$; let $w^{\prime}$ the white node
below it in $I_{3}$. Comb strands left of this pair $bw$ as in Figure 25,
left, so there are no crossings left of strand $wb$.

Figure 25: Left panel: combing the strand from $w$ to $b$. Right panel: after
applying the basic skein relation to the two crossings near $b$.

Apply skein relations to the strand crossings between $bw$ and the strand from
$w^{\prime}$ and the strand from the black vertex below it, as in the Figure,
right panel. In order to obtain $\lambda$, the thick edges of Figure 25,
right, must be part of a quad face in some subsequent skein reduction
(otherwise the outer face between $b$ and $w^{\prime}$ would remain of degree
$3$). Either reduction of this quad face results in a web which cannot be
reduced to $\lambda$. If $b$ happens to be the lower black vertex of $I_{3}$,
$w$ can not be the left-most white node of $I_{1}$ (otherwise ${\cal
P}_{\lambda,\tau}=0$) and then the same argument using the last two crossings
near $w$ (with the roles of $b$ and $w$ switched) applies.

If for each $i$ every white node in $I_{i}$ connects to a black node in
$I_{i+1}$, then we claim ${\cal P}_{\lambda,\tau}=(-1)^{n}$. As in the
previous proof, consider first crossings between pairs connecting white nodes
in $I_{i}$ to black nodes in $I_{i+1}$. Any such crossing must resolve into
parallel strands. Resolving all these crossings leads to the pairing
$\tau_{\text{cross}}$ of Figure 26. Each crossing of $\tau_{\text{cross}}$ (of
which there are $3n^{2}$) must then resolve to a double-Y, and thus ${\cal
P}_{\lambda,\tau}={\cal
P}_{\lambda,\tau_{\text{cross}}}=(-1)^{3n^{2}}=(-1)^{n}$. This implies the
statement.

Figure 26: Basic crossing diagram for the order-$3$ honeycomb web.

∎

For boundary colors $c\equiv 1$, the partition function is
$Z(c)=Z(Z_{int})^{2}=\Delta^{3}\det X$, which is the denominator in the
expression for the probability of $T_{n}$. We thus have, for the measure
$\mu_{c}$,

$Pr(T_{n})=\frac{P_{T_{n}}\mathrm{Tr}_{T_{n}}(1)}{\det X}.$ (25)

Although we don’t have an exact expression for $\mathrm{Tr}_{T_{n}}$, Baxter
[Bax70] computed the asymptotic growth rate of the trace of the honeycomb web
to be

$\lim_{n\to\infty}\frac{1}{n^{2}}\log
Tr_{T_{n}}(1)=\frac{1}{2}\log(\frac{3\Gamma(\frac{1}{3})^{3}}{4\pi^{2}}).$

###  7.2 The triangular honeycomb $T_{n}^{\prime}$

For $T_{n}^{\prime}$, all $3n$ nodes are white. They are grouped into
subintervals $W^{1}=\\{1,\dots,n\\}$, $W^{2}=\\{n+1,\dots,2n\\}$ and
$W^{3}=\\{2n+1,\dots,3n\\}$. The matrix $X$ is $3n\times n$. Let $B$ denote
the columns of $X$. Let $\det X_{W^{i}}^{B}$ be the determinant of the maximal
minor of $X$ corresponding to rows $W^{i}$.

######  Theorem 9.

We have $P_{T^{\prime}_{n}}=\det X_{W^{1}}^{B}\det X_{W^{2}}^{B}\det
X_{W^{3}}^{B}.$

Figure 27: Triangular honeycomb web $T^{\prime}_{n}$ of order $4$.

###### Proof.

Let $\lambda=T_{n}^{\prime}$. By Theorem 4, we need to find $3$-partitions
$\tau$ (partitions into parts of size $3$) of $\\{1,\dots,3n\\}$ which
contribute to ${\cal P}_{\lambda,\tau}.$ We prove that $\tau$ contributes if
and only if each triple of $\tau$ contains one element of each of
$W^{1},W^{2},W^{3}$.

First suppose a part of $\tau$ contains two nodes $w,w^{\prime}$ of the same
subinterval, say $W^{1}$. We then claim that ${\cal P}_{\lambda,\tau}=0$. We
apply skein relations to write $\tau$ as a formal linear combination of planar
reduced webs. We argue by induction on the number of nodes separating $w$ and
$w^{\prime}$. If $w,w^{\prime}$ are adjacent, and connected to internal vertex
$b$, then in any planar resolution $w,b,w^{\prime}$ will be connected in the
same way $w\sim b\sim w^{\prime}$, that is, no skein relation changes these
connections. Thus ${\cal P}_{\lambda,\tau}=0$. If $w,w^{\prime}$ are not
adjacent, first “comb” the crossings so that there are no crossing of strands
between $w,b,w^{\prime}$ and the boundary, as in Figure 28.

Figure 28:  Combing of $\tau$ between $w$ and $w^{\prime}$.

Let $w_{1}$ be the first node adjacent to $w$, between $w$ and $w^{\prime}$.
Now either resolution of the crossing of the strand starting at $w_{1}$ and
the path $wbw^{\prime}$ results in a web with two connected nodes closer then
$w,w^{\prime}$ are. The completes the proof that ${\cal P}_{\lambda,\tau}\neq
0$ only if each part of $\tau$ contains nodes from each of
$W^{1},W^{2},W^{3}$.

Now suppose each part of $\tau$ contains a point in each of
$W^{1},W^{2},W^{3}$. We show that ${\cal P}_{\lambda,\tau}=\pm 1$ with the
sign given by the above determinantal formula. We show this by induction on
$n$, the number of parts of $\tau$. If $n=1$, the conclusion holds. Suppose we
have constructed a copy of $T_{n-1}^{\prime}$ from the first $n-1$ parts of
$\tau$. We embed the last part $ijk$ of $\tau$ so that its central vertex $b$
is near the lower left corner of the triangle. This means the strand from $i$
to $b$ crosses $i-1$ vertical strands on the lower level of
$T_{n-1}^{\prime}$, the strand from $j$ crosses $j-n-1$ strands on side $2$
and $n-1$ strands on side $1$, and the strand from $k$ crosses $3n-k$ strands
on side $3$. See Figure 29.

Figure 29: Induction step.

These crossings all necessarily resolve as parallel crossings except for the
$n$ crossings of strand $j$ along side $1$, which result in $n-2$ more
hexagonal faces and thus creates $T_{n}^{\prime}$.

To compute the sign contribution, take a $3$-partition $\tau$ and bijection
$\sigma$ from the parts of $\tau$ to $B_{int}^{*}$. Compare $\tau$ with the
partition $\tau_{0}$ into parts $(i,n+i,2n+i)$ with $i$ from $1$ to $n-1$, and
bijection sending $(i,n+i,2n+i)$ to $b_{i}$, the $i$th element of
$B_{int}^{*}$. We can order the parts of $\tau$ according to $\sigma$. Then
$\tau$ is defined by $3$ permutations $\sigma_{1},\sigma_{2},\sigma_{3}$ where
$\sigma_{i}$ orders the parts in $W^{i}$. Now $\tau$ can be drawn in the
triangle so that it agrees with the drawing of $\tau_{0}$ except for a narrow
band along each edge where the strands cross according to the permutation
$\sigma_{i}$. We can reduce $\tau$ by first uncrossing all the strands in the
bands to make a copy of $\tau_{0}$, then reducing $\tau_{0}$. Thus the sign
from the contribution of $\tau$ equals
$(-1)^{\sigma_{1}+\sigma_{2}+\sigma_{3}}$ times the sign of the contribution
of $\tau_{0}$.

This is the same sign as that given by the product of determinants in the
statement. ∎

##  8 Scaling limit

Let $\mathbb{H}$ denote the upper half plane and suppose
${\mathcal{G}}_{\varepsilon}=\varepsilon{\mathbb{Z}}^{2}\cap\mathbb{H},$ the
part of $\varepsilon{\mathbb{Z}}^{2}$ in the closed upper half plane. Let
$z_{1},z_{2}\in{\mathbb{R}}$ be distinct points on the boundary of
$\mathbb{H}$. In ${\mathcal{G}}_{\varepsilon}$ let
$w_{\varepsilon},b_{\varepsilon}$ be, respectively, a white and black vertex
on the lower boundary of ${\mathcal{G}}_{\varepsilon}$, and converging to
$z_{1},z_{2}$ as $\varepsilon\to 0$. We have

$X_{w_{\varepsilon},b_{\varepsilon}}=\frac{2\varepsilon}{\pi(z_{2}-z_{1})}+O(\varepsilon^{2}),$
(26)

see [Ken00].

Let $w_{1}<b_{1}<w_{2}<b_{2}$ be four points in ${\mathbb{R}}$. Let
$w_{1}^{\varepsilon},b_{1}^{\varepsilon},w_{2}^{\varepsilon},b_{2}^{\varepsilon}$
be white and black vertices of ${\mathcal{G}}_{\varepsilon}$ converging to the
$w_{i},b_{i}$ as above. Then with $c\equiv 1$ a random reduced web will
connect $w_{1}$ with $b_{1}$ and $w_{2}$ with $b_{2}$ with probability (see
(16) above)

$\Pr(\lambda_{1})=\frac{X_{1,1}X_{2,2}}{X_{1,1}X_{2,2}-X_{1,2}X_{2,1}}~{}~{}\stackrel{{\scriptstyle\varepsilon\to
0}}{{\longrightarrow}}~{}~{}\frac{(b_{2}-w_{1})(w_{2}-b_{1})}{(b_{2}-b_{1})(w_{2}-w_{1})}.$

Note that this limiting quantity is Möbius invariant; it only depends on the
cross ratio of the four boundary points. More generally this quantity is
conformally invariant, if we replace the upper half plane with a Jordan domain
with four marked boundary points, approximated by
$\varepsilon{\mathbb{Z}}^{2}$ as $\varepsilon\to 0$ in an appropriate sense
[Ken00].

Let $w_{1}<b_{1}<\dots<w_{3n}<b_{3n}$ be $6n$ points on the boundary of
$\mathbb{H}$. Let
$w_{1}^{\varepsilon},b_{1}^{\varepsilon},\dots,w_{3n}^{\varepsilon},b_{3n}^{\varepsilon}$
be white and black vertices of ${\mathcal{G}}_{\varepsilon}$ converging to the
$w_{i},b_{i}$ as above. For boundary colors $c\equiv 1$ the probability of the
honeycomb web $T_{n}$ of Figure 24, given by expression (25), takes a nice
form in the $\varepsilon\to 0$ limit since the determinants are of Cauchy
matrices. Letting $I_{1}=\\{1,\dots,n\\},~{}~{}I_{2}=\\{n+1,\dots,2n\\}$ and
$I_{3}=\\{2n+1,\dots,3n\\}$,

$Pr(T_{n})\to\frac{\prod_{\text{$(i,j)$ not in $(I_{1},I_{2}),(I_{2},I_{3})$
or $(I_{3},I_{1})$}}(w_{i}-b_{j})}{\prod_{\text{$i,j$ in diff't
parts}}(w_{i}-w_{j})(b_{i}-b_{j})}\mathrm{Tr}_{T_{n}}(1).$ (27)

See for example (1). This case is special in that the numerators and
denominators factor into linear parts; this factorization does not hold for
more general boundary conditions.

##  9 Appendix: $n=6$ cases

When $n=6$, with three white and three black nodes there are, up to symmetry,
three possible type patterns. For each pattern, $\Lambda$ consists in six
classes. The other case with $n=6$ consists of all nodes of the same type.

###  9.1 Type pattern $(w,b,w,b,w,b)$

When $p=(w,b,w,b,w,b)$, the six reduced web classes are shown in Figure 30.

Figure 30: Classes of reduced webs with $6$ boundary points and
$p=(w,b,w,b,w,b)$.

The matrix ${\cal P}$ is (rows are indexed by the webs in the figure; column
labels correspond to the permutation from white to black, in line notation,
defined by pairing $\tau$.

123 132 213 231 312 321 $1$ $1$ $0$ $0$ $0$ $0$ $0$ $2$ $0$ $0$ $0$ $0$ $1$
$0$ $3$ $0$ $1$ $0$ $1$ $0$ $0$ $4$ $0$ $0$ $1$ $1$ $0$ $0$ $5$ $0$ $0$ $0$
$1$ $0$ $1$ $6$ $0$ $0$ $0$ $-1$ $0$ $0$

The corresponding polynomials are (recall that the coefficients in the table
need to be multiplied by the signature of $\tau$)

$\begin{pmatrix}P_{1}\\\ \vdots\\\
P_{6}\end{pmatrix}=\begin{pmatrix}X_{1,1}X_{2,2}X_{3,3}\\\
X_{1,3}X_{2,1}X_{3,2}\\\ -X_{1,1}X_{2,3}X_{3,2}+X_{1,2}X_{2,3}X_{3,1}\\\
-X_{1,2}X_{2,1}X_{3,3}+X_{1,2}X_{2,3}X_{3,1}\\\
-X_{1,3}X_{2,2}X_{3,1}+X_{1,2}X_{2,3}X_{3,1}\\\
-X_{1,2}X_{2,3}X_{3,1}\end{pmatrix}.$

If $c\equiv 1$, note that (after multiplying the last one by $2$, its trace)
their sum is $\det X=\frac{Z(c)}{\Delta^{3}}$.

In the scaling limit with (26), and $w_{1},b_{1},w_{2},b_{2},w_{3},b_{3}\to
z_{1},z_{2},z_{3},z_{4},z_{5},z_{6}$ we have limiting probabilities

$\begin{pmatrix}Pr_{1}\\\ \vdots\\\
Pr_{6}\end{pmatrix}=\begin{pmatrix}\frac{(z_{3}-z_{2})(z_{4}-z_{1})(z_{5}-z_{2})(z_{5}-z_{4})(z_{6}-z_{1})(z_{6}-z_{3})}{(z_{3}-z_{1})(z_{4}-z_{2})(z_{5}-z_{3})(z_{6}-z_{4})(z_{5}-z_{1})(z_{6}-z_{2})}\\\
\frac{(z_{2}-z_{1})(z_{4}-z_{1})(z_{4}-z_{3})(z_{5}-z_{2})(z_{6}-z_{3})(z_{6}-z_{5})}{(z_{3}-z_{1})(z_{4}-z_{2})(z_{5}-z_{3})(z_{6}-z_{4})(z_{5}-z_{1})(z_{6}-z_{2})}\\\
\frac{(z_{3}-z_{2})(z_{4}-z_{3})(z_{6}-z_{1})(z_{6}-z_{5})}{(z_{3}-z_{1})(z_{5}-z_{3})(z_{6}-z_{4})(z_{6}-z_{2})}\\\
\frac{(z_{2}-z_{1})(z_{4}-z_{3})(z_{5}-z_{4})(z_{6}-z_{1})}{(z_{3}-z_{1})(z_{4}-z_{2})(z_{6}-z_{4})(z_{5}-z_{1})}\\\
\frac{(z_{2}-z_{1})(z_{3}-z_{2})(z_{5}-z_{4})(z_{6}-z_{5})}{(z_{4}-z_{2})(z_{5}-z_{3})(z_{6}-z_{2})(z_{5}-z_{1})}\\\
\frac{2(z_{2}-z_{1})(z_{3}-z_{2})(z_{4}-z_{3})(z_{5}-z_{4})(z_{6}-z_{5})(z_{6}-z_{1})}{(z_{3}-z_{1})(z_{4}-z_{2})(z_{5}-z_{3})(z_{6}-z_{4})(z_{5}-z_{1})(z_{6}-z_{2})}\end{pmatrix}$
(28)

This case is special in that the numerators and denominators factor into
linear parts; this factorization does not hold for more general boundary
conditions.

###  9.2 Type pattern $(b,b,w,b,w,w)$

When $p=(b,b,w,b,w,w)$, the six reduced web classes are shown in Figure 31.

Figure 31: Classes of reduced webs with $6$ boundary points and
$p=(b,b,w,b,w,w)$.

The matrix ${\cal P}$ is

123 132 213 231 312 321 $1$ $0$ $0$ $0$ $0$ $1$ $1$ $2$ $1$ $1$ $1$ $1$ $0$
$0$ $3$ $0$ $0$ $0$ $0$ $-1$ $0$ $4$ $-1$ $0$ $-1$ $0$ $0$ $0$ $5$ $-1$ $-1$
$0$ $0$ $0$ $0$ $6$ $1$ $0$ $0$ $0$ $0$ $0$

Consequently the polynomials are

$\begin{pmatrix}P_{1}\\\ \vdots\\\
P_{6}\end{pmatrix}=\begin{pmatrix}X_{1,3}X_{2,1}X_{3,2}-X_{1,3}X_{2,2}X_{3,1}\\\
X_{1,1}X_{2,2}X_{3,3}-X_{1,1}X_{2,3}X_{3,2}-X_{1,2}X_{2,1}X_{3,3}+X_{1,2}X_{2,3}X_{3,1}\\\
-X_{1,3}X_{2,1}X_{3,2}\\\ -X_{1,1}X_{2,2}X_{3,3}+X_{1,2}X_{2,1}X_{3,3}\\\
-X_{1,1}X_{2,2}X_{3,3}+X_{1,1}X_{2,3}X_{3,2}\\\
X_{1,1}X_{2,2}X_{3,3}\end{pmatrix}.$

###  9.3 Type pattern $(b,b,b,w,w,w)$

This case is shown in Figure 32.

Figure 32: Classes of reduced webs with $6$ boundary points and
$p=(b,b,b,w,w,w)$.

The matrix ${\cal P}$ is

123 132 213 231 312 321 $1$ $1$ $1$ $1$ $1$ $1$ $1$ $2$ $-1$ $-1$ $-1$ $0$
$-1$ $0$ $3$ $-1$ $-1$ $-1$ $-1$ $0$ $0$ $4$ $-1$ $0$ $0$ $0$ $0$ $0$ $5$ $1$
$1$ $0$ $0$ $0$ $0$ $6$ $1$ $0$ $1$ $0$ $0$ $0$

###  9.4 Type pattern $(w,w,w,w,w,w)$

In this case there are five classes, see Figure 33.

Figure 33: Classes of reduced webs with $6$ boundary points and
$p=(w,w,w,w,w,w)$.

The matrix ${\cal P}$ is (with columns indexing the 3-partitions, enumerated
by the part of the partition containing $1$)

123 124 125 126 134 135 136 145 146 156 $1$ $0$ $0$ $1$ $1$ $0$ $1$ $1$ $0$
$0$ $0$ $2$ $1$ $1$ $0$ $0$ $0$ $1$ $0$ $1$ $0$ $0$ $3$ $0$ $0$ $0$ $0$ $1$
$1$ $0$ $0$ $1$ $1$ $4$ $0$ $0$ $0$ $0$ $0$ $-1$ $-1$ $-1$ $-1$ $0$ $5$ $0$
$-1$ $-1$ $0$ $-1$ $-1$ $0$ $0$ $0$ $0$

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