2.1. Ribbon graphs and arrow presentations

Our interest is in polynomials of graphs cellularly embedded in closed surfaces. It is usual in the area to describe these as ribbon graphs.

A ribbon graph $\mathbb{G}=\left(V,E\right)$ is a surface with boundary, represented as the union of two sets of discs — a set V of vertices and a set E of edges — such that: (1) the vertices and edges intersect in disjoint line segments; (2) each such line segment lies on the boundary of precisely one vertex and precisely one edge; and (3) every edge contains exactly two such line segments. A ribbon graph is shown in Figure 2a). Two ribbon graphs $\mathbb{G}$ and $\mathbb{G}'$ are equivalent if is there is a homeomorphism from $\mathbb{G}$ to $\mathbb{G}'$ (orientation preserving when $\mathbb{G}$ is orientable) mapping vertices to vertices and edges to edges. In particular, the homeomorphism preserves the cyclic order of half-edges at each vertex. Note that as a ribbon graph is topologically a surface with boundary, it has a unique embedding into a closed surface. A ribbon graph is plane if, topologically, it is a collection of spheres with holes.

One thing that can be tricky when working with ribbon graphs is that vertices can change in dramatic ways when contracting an edge that is a loop (i.e., an edge incident to only one vertex). This can result in loop and non-loop edges needing to be treated separately. However, if we describe ribbon graphs by “arrow presentations”, this distinction is not necessary.

An arrow presentation $\mathbb{G}$ consists of a set of circles (i.e., closed 1-manifolds) and a set of labels. For each label there are exactly two arrows lying along one or more of the circles. All arrows are disjoint. The set of labels is called the edge set and its elements are edges. By an arc in an arrow presentation we mean a subset of a circle that is homeomorphic to a closed interval. In general, an arc may or may not contain an arrow.

Figure 2b) shows an arrow presentation. It corresponds to the ribbon graph shown in Figure 2a). It is important to note that the circles of an arrow presentation are not regarded as being drawn or embedded in the plane. For example, see the arrow presentation illustrated in Figure 3b).

Arrow presentations describe ribbon graphs as follows: if $\mathbb{G}$ is a ribbon graph, for each edge e, arbitrarily orient the boundary of that edge, place an e-labelled arrow on each of the arcs where e intersects a vertex, pointing in the direction given by the edge’s boundary orientation. Taking the boundaries of the vertices together with the labelled arrows gives an arrow presentation. On the other hand, given an arrow presentation, obtain a ribbon graph by identifying each circle with the boundary of a vertex disk. Then, for each label e, take a disc with an orientation of its boundary and identify an arc on its boundary with an e-labelled arrow such that the direction of each arrow agrees with the orientation. Thus, in an arrow presentation, each circle corresponds to a vertex of a ribbon graph, and each pair of e-labelled arrows corresponds to an edge. We say that two arrow presentations are equivalent if they describe the same ribbon graph. This happens when you can move from one arrow presentation to the other through renaming edges and reversing the direction of all arrows labelled by some subset of edges. Arrow presentations are considered up to equivalence.

The following operations are illustrated in Figure 1. If $\mathbb{G}$ is an arrow presentation and e is one of its edges (i.e., labels), then an arrow presentation for the deletion $\mathbb{G}\backslash e$ is obtained by removing the label e together with its two arrows (leaving the circles intact). The partial dual, introduced in [Reference Chmutov8], $\mathbb{G}^{\{e\}}$ of $\mathbb{G}$ is the arrow presentation obtained as follows. Suppose a and b are the two e-labelled arrows, with heads $h(a),h(b)$ and tails $t(a), t(b)$, respectively. Place an arrow from h(b) to t(a), and place another arrow from h(a) to t(b). Label both new arrows with e. Delete a, b and the arcs of the circles (or circle) on which they lay. Note that this changes the set of circles. The contraction $\mathbb{G}/ e$ is defined as $\mathbb{G}^{\{e\}}\backslash e$. The partial Petrial $\mathbb{G}^{\tau(e)}$ is obtained by reversing the direction of exactly one of the e-labelled arrows. If A is a set of edges then $\mathbb{G}\backslash A$, $\mathbb{G}^{A}$, $\mathbb{G}/ A$, and $\mathbb{G}^{\tau(A)}$ are all defined by applying the relevant operation to each edge in A in any order. (The result is obviously independent of the choice of order.)

Figure 1. Operations on arrow presentations.

We say an arrow presentation $\mathbb{K}$ is a sub-arrow presentation of $\mathbb{G}$ if it can be obtained from $\mathbb{G}$ by deleting some number of edges and/or circles. A sub-arrow presentation is induced by an edge set A if it consists, exactly, of the arrows with labels in A together with any circles on which they lie.

We shall make use of the following parameters of arrow presentations. They are direct translations of the corresponding ribbon graph terms. If E is the set of edges of an arrow presentation $\mathbb{G}$ and $A\subseteq E$ then:

• • $e(A)=|A|$;

• • v(A) is the number of circles in $\mathbb{G}\backslash(E\setminus A)$ (which is the same as the number of circles in $\mathbb{G}$, but we use v(A) for clarity in its usage);

• • k(A) is the number of equivalence classes of circles in $\mathbb{G}\backslash(E\setminus A)$ where two circles are equivalent if they each contain an arrow of the same label (this equals the number of connected components of the ribbon graph described by $\mathbb{G}\backslash(E\setminus A)$);

• • $r(A)=v(A)-k(A)$ is the rank of A;

• • b(A) is the number of circles in $\mathbb{G}/A $ or equivalently in $\mathbb{G}\backslash(E\setminus A)/A$ (this is the number of boundary components of the ribbon graph described by $\mathbb{G}\backslash(E\setminus A)$);

• • $\gamma(A)$ $=2k(A)-v(A)+e(A)-b(A)$ is the Euler genus of $\mathbb{G}\backslash(E\setminus A)$, the equation being Euler’s formula.

If the ribbon graph which the arrow presentation $\mathbb{G}\backslash (E\setminus A)$ describes is non-orientable then $\gamma(E)$ gives its genus, and if it is orientable, twice its genus. The arrow presentation $\mathbb{G}$ is plane if $\gamma(E)$=0, and represents a connected ribbon graph if $k(E)=1$.

2.2. Ribbon graph polynomials

For context we begin with the Tutte polynomial of a graph $G=(V,E)$, which can be defined by

(2.1)\begin{equation} T(G;x,y) := \sum_{A\subseteq E} (x-1)^{r(E)-r(A)} (y-1)^{|A|-r(A)}, \end{equation}

where r(A) is the rank of the subgraph (V, A) of G.

The analogue of the Tutte polynomial for an arrow presentation (or ribbon graph) $\mathbb{G}$ is the ribbon graph polynomial, $R(\mathbb{G};x,y)$, which is a universal deletion-contraction invariant for ribbon graphs. Its definition differs from (2.1) by modifying the rank function so that it records some topological information about the embedding. Let $\rho(A):= r(A)+\tfrac{1}{2}\gamma (A)$. Then the ribbon graph polynomial of an arrow presentation $\mathbb{G}$ on edge set E is

(2.2)\begin{equation} R(\mathbb{G};x,y) := \sum_{A\subseteq E} (x-1)^{\rho(E)-\rho(A)} (y-1)^{|A|-\rho(A)}. \end{equation}

When $\mathbb{G}$ is plane $R(\mathbb{G};x,y)=T(\mathbb{G};x,y)$ although these polynomials do not agree in general. We note that the polynomial $R(\mathbb{G};x,y)$ is, up to a normalization, a two-variable specialization of the well known four-variable Bollobás–Riordan polynomial of [Reference Bollobás and Riordan4].

The topological transition polynomial, introduced in [Reference Ellis-Monaghan and Moffatt11], contains the ribbon graph polynomial as well as the Penrose polynomial of [Reference Aigner1, Reference Ellis-Monaghan and Moffatt13] as specializations (see [Reference Ellis-Monaghan and Moffatt12] for details). It is intimately related to Jaeger’s transition polynomial [Reference Jaeger17] and the generalized transition polynomial of [Reference Ellis-Monaghan and Sarmiento15]. Here we consider the four-variable version of the polynomial, although a multivariate version is often used. (The particular form of the polynomial we use here can be found in [Reference Ellis-Monaghan and Moffatt14].)

The topological transition polynomial, $Q(\mathbb{G}; (a, b, c) , t)$, of an arrow presentation $\mathbb{G}$ on an edge set E can be defined as

\begin{equation*}Q(\mathbb{G}; (a,b,c) , t) :=\sum_{(X,Y,Z) \in \mathcal{P}_3(E)} a^{|X|} b^{|Y|} c^{|Z|} t^{b( \mathbb{G}^{\tau(Z)}\backslash Y)},\end{equation*}

where $\mathcal{P}_3(E)$ denotes the set of ordered partitions of E into three blocks, which may be empty. As shown in [Reference Ellis-Monaghan and Moffatt11], the topological transition polynomial can be defined by the recursion relation

(2.3)\begin{equation} Q(\mathbb{G})=a \, Q(\mathbb{G}\operatorname{/} e)+b \,Q( \mathbb{G}\backslash e)+c\, Q(\mathbb{G}^{\tau(e)} \operatorname{/} e),\end{equation}

together with its value of t ^v when $\mathbb{G}$ is an edgeless arrow presentation on v circles.

3.1. Separations of arrow presentations and ribbon graphs

We start by describing a particular decomposition of an arrow presentation into two sub-arrow presentations, or equivalently a ribbon graph into two ribbon subgraphs.

Let $\mathbb{G}$ be an arrow presentation with edge set E. Suppose that H and K are two sets of edges that partition E. Let $\mathbb{H}$ be the arrow presentation induced by H, and $\mathbb{K}$ be the arrow presentation induced by K. If $\mathbb{G}$ has any isolated circles (i.e., circles without arrows) then include each one in either of $\mathbb{H}$ or $\mathbb{K}$.

An example is shown in Figure 2, where Figures 2c) and 2d) show two different 2-separations of the arrow presentation of Figure 2b). The welding-arcs are indicated in blue (light grey if viewed in greyscale). The welding-arcs have been shortened for clarity, a convention that we shall follow in all our figures. Properly, a welding-arc consists of the whole of the arc between the arrows.

Figure 2. An arrow presentation, its corresponding ribbon graph, and two 2-separations. (a) A ribbon graph. (b) An arrow presentation. (c) A 2-separation. (d) Another 2-separation.

Note that in general the welding-arcs in an r-separation may lie on fewer than r circles.

Suppose an arrow presentation $\mathbb{G}$ has an r-separation defined by some set of welding-arcs, and that $\mathbb{G}'$ is obtained from $\mathbb{G}$ by applying deletion, contraction, partial duality, and partial Petriality to some edges. Then the welding-arcs of $\mathbb{G}$ induce a set of arcs on $\mathbb{G}'$. We also call these arcs welding-arcs. These welding-arcs may or may not define an r-separation (as they may not lie between arrows). Later we shall make use of the following instance where they do.

We will be interested in the arrow presentations that result from applying the deletion-contraction operations to all of the edges of $\mathbb{G}$ that belong to $\mathbb{H}$. To discuss these we set up the following notation. We shall make extensive use of this notation ( $\mathbb{H}[\kappa]$ and $\mathbb{K}[\eta]$ ) in what follows.

The reader may find it helpful to consult the example in Figure 3 while reading the following definition of $\mathbb{H}[\kappa]$. For this example, $\mathbb{G}$ is shown in Figure 3a), $\mathbb{H}$ has edge set $\{a,b,c,d\}$, $\mathbb{K}$ has edge set $\{x,y,z\}$, and the welding-arcs are numbered. Then, Figure 3b) shows $\mathbb{H}[\kappa]$ where κ is the pairing $\{\{1,4\},\{2,5\},\{3,6\}\}$.

Figure 3. An example of an arrow presentation, a 3-separation, and $\mathbb{H}[\kappa]$. (a) $\mathbb{G}$. (b) $\mathbb{H}[\{\{1,4\},\{2,5\},\{3,6\}\}]$.

Note that $\langle \eta,\kappa\rangle$ can be obtained as $(\mathbb{H}[\kappa])[\eta]$ or $(\mathbb{K}[\eta])[\kappa]$.

This indexing provides orderings on the sets of pairings. We do not assume any relation between the two orderings.

3.2. The splitting formula

Our aim is to find a splitting formula for the ribbon graph polynomial. Rather than working with the polynomial $R(\mathbb{G};x,y)$ of Equation (2.2), it is convenient to work with a different but equivalent polynomial. We define a polynomial $P(\mathbb{G}; x,y,t)$ through the deletion-contraction relations

(3.1)\begin{equation} P(\mathbb{G}):= \begin{cases} xP({\mathbb G}/e)+yP({\mathbb G}\backslash e), & \text{where}\ e\ \text{is any edge of } \mathbb{G}; \\ t^{v(E)} & \text{when}\ \mathbb{G}\ \text{is edgeless with}\ v(E)\ \text{circles} . \end{cases} \end{equation}

For simplicity we have written $P(\mathbb{G})$ for $P(\mathbb{G}; x,y,t)$, etc., in the above. It is routine to check that

\begin{equation*}P(\mathbb{G}; x,y,t)=\sum_{A\subseteq E}x^{|A|}y^{|E\setminus A|}t^{b(A)},\end{equation*}

and from this that $P(\mathbb{G})$ is well-defined, and that $P(\mathbb{G})$ and $R(\mathbb{G})$ are equivalent:

(3.2)\begin{equation} R(\mathbb{G};x+1,y+1) = x^{-b(E)/2} y^{-v(E)/2}\, P(\mathbb{G}; y^{1/2},x^{1/2},(xy)^{1/2}), \end{equation}

and

(3.3)\begin{equation} P(\mathbb{G},x,y,t) = (xy/t)^{e(E)/2} (ty/x)^{b(E)/2} (xt/y)^{v(E)/2} R(\mathbb{G};ty/x+1, xt/y+1). \end{equation}

Remark. When $\mathbb{G}$ is plane, it follows by Euler’s formula and [Reference Negami21, Theorem 1.3] that $P(\mathbb{G})$ is equivalent to Negami’s polynomial:

(3.4)\begin{equation}t^{v(E)}P(\mathbb{G},x,y,t) = f(\mathbb{G};xt,y,t^2), \end{equation}

and

(3.5)\begin{equation} f(\mathbb{G};x,y,t)= t^{v(E)/2}P(\mathbb{G};x/t^{1/2}, y, t^{1/2}).\end{equation}

When comparing [Reference Negami21, Equations (i) and (ii)] and Equation (3.1) $P(\mathbb{G})$ and $f(\mathbb{G})$ may appear to have identical deletion-contraction relations, however the contraction operations appearing in them do not act in the same way.

One reason for working with $P(\mathbb{G})$ rather than $R(\mathbb{G})$ is that it has simpler deletion-contraction relations than $R(\mathbb{G})$ (which can be found written down in [Reference Ellis-Monaghan and Moffatt12, Corollary 4.42] or [Reference Huggett and Moffatt16, Equation (4.1)]). But also the fact that $P(\mathbb{G})$ takes the value of $t^{v(E)}$, rather than 1, on edgeless arrow presentations is crucial for our arguments.

We now state and prove our main result. An example is provided at the end of this subsection.

Theorem 1. Let $\mathbb{G}$ be an arrow presentation that is r-separated into $\mathbb{H}$ and $\mathbb{K}$. The r-separation is defined by 2r welding-arcs. Let $\{\eta_{i}:i=1,\dots d\}$ be an ordering of the set of pairings of $\mathbb{H}$-ends of these arcs, and $\{\kappa_{j}:j=1,\dots d\}$ be an ordering of the set of pairings of $\mathbb{K}$-ends. Then

(3.6)\begin{equation} P(\mathbb{G}) = \Big[P(\mathbb{K}[\eta_i])\Big]^T \cdot \Big[t^{c\langle \eta_{i},\kappa_{j}\rangle} \Big]^{-1} \cdot \Big[ P(\mathbb{H}[\kappa_j] )\Big] ,\end{equation}

where the square brackets denote matrices or vectors in the obvious way.

Proof. Arbitrarily order the edges of $\mathbb{G}$ ensuring that the edges of $\mathbb{H}$ precede the edges of $\mathbb{K}$. Throughout the proof we will assume that the deletion-contraction relations are applied with respect to this order, smallest first.

Consider the sub-arrow presentation $\mathbb{H}$. By applying the deletion-contraction relation (3.1) to all of the edges in $\mathbb{H}$ we can write

(3.7)\begin{equation} P(\mathbb{H}) = \sum_{l\in \{1,\ldots, 2^{|E(\mathbb{H})|}\}} w_l(x,y) \,P(\mathbb{L}_l), \end{equation}

where $w_l(x,y)$ is some monomial in x and y and $\mathbb{L}_l$ is some arrow presentation.

Each arrow presentation $\mathbb{L}_l$ consists of a set of arrowless circles, some of which contain welding-arcs (inherited from $\mathbb{G}$). Those that contain welding-arcs are of the form $\langle \eta_i, \kappa(\mathbb{K})\rangle$ where $\kappa(\mathbb{K})$ is the pairing of the $\mathbb{K}$-ends given by following the arcs of the circles of the arrow presentation $\mathbb{G}$ that lie between two $\mathbb{K}$-ends and do not include an $\mathbb{H}$-end. Thus

(3.8)\begin{equation} P(\mathbb{L}_l) = t^{r_l} t^{c\langle \eta_i, \kappa(\mathbb{K})\rangle } \end{equation}

for some r_l and η_i. By collecting together terms we may therefore write Equation (3.7) as

(3.9)\begin{equation} P(\mathbb{H}) = \sum_{\eta_i} A(\eta_i, \kappa(\mathbb{K})) \; t^{c\langle \eta_i, \kappa(\mathbb{K})\rangle }. \end{equation}

for some polynomials $A(\eta_i, \kappa(\mathbb{K}))$ in $x,y,t$ that are completely determined by the $t^{r_l}$ and $w_l(x,y)$. (Some of the $A(\eta_i, \kappa(\mathbb{K}))$ may be zero.) This equation defines the $A(\eta_i, \kappa(\mathbb{K}))$ for the rest of the proof.

If instead of $\mathbb{H}$ we started the above process with $\mathbb{H}[\kappa_j]$ for some κ_j, since all that changes is the terminal forms resulting from the deletion-contraction, we see that

(3.10)\begin{equation} P(\mathbb{H}[\kappa_j]) = \sum_{\eta_i} A(\eta_i, \kappa(\mathbb{K})) \; t^{c\langle \eta_i, \kappa_j\rangle }. \end{equation}

where the $A(\eta_i, \kappa(\mathbb{K}))$ are as above. (The $A(\eta_i, \kappa(\mathbb{K}))$ are unchanged since the factors of t in this expression come from the circles that do not involve welding-arcs.)

Next we repeat this process for the arrow presentation $\mathbb{G}$. By applying the deletion-contraction relation (3.1) to all of the edges in $\mathbb{H}$ but none of the edges in $\mathbb{K}$

(3.11)\begin{equation} P(\mathbb{G}) = \sum_{l\in \{1,\ldots, 2^{|E(\mathbb{H})|}\}} w_l(x,y) \,P(\mathbb{L}'_l), \end{equation}

where the $w_l(x,y)$ are exactly those in Equation (3.7) and the $\mathbb{L}'_l$ are arrow presentations that only contain arrows of $\mathbb{K}$.

As we have applied deletion-contraction in the same way to obtain each of Equations (3.7) and (3.11), each $\mathbb{L}'_l$ naturally corresponds with some $\mathbb{L}_l$. (This correspondence was implicitly assumed when re-using the $w_l(x,y)$ in Equation (3.11).) Under this correspondence, each $\mathbb{L}_l$ can be obtained from $\mathbb{L}'_l$ by forgetting all of the arrows and removing any circles that were vertices of $\mathbb{K}$ but not $\mathbb{H}$. From Equation (3.8) we have $P(\mathbb{L}_l) =t^{r_l} t^{c\langle \eta_i,\kappa(\mathbb{K})\rangle}$. From our correspondence we now have that $P(\mathbb{L}_l') = t^{r_l} P( \mathbb{K}[\eta_i])$ for the same r_l and η_i. Then collecting together terms as we did for Equation (3.9), we see

(3.12)\begin{equation} P(\mathbb{G}) = \sum_{\eta_i} A(\eta_i, \kappa(\mathbb{K})) \; P(\mathbb{K}[\eta_i]). \end{equation}

Finally, from Equation (3.10) we obtain the system of linear equations

\begin{equation*} \Big[ P(\mathbb{H}[\kappa_j] )\Big] = \Big[t^{c\langle \eta_{i},\kappa_{j}\rangle} \Big] \cdot \Big[ A(\eta_i, \kappa(\mathbb{K})) \Big]. \end{equation*}

But by Lemma 1 the matrix $\Big[t^{c\langle \eta_{i},\kappa_{j}\rangle} \Big]$ is invertible. So

\begin{equation*}P(\mathbb{G}) = \Big[P(\mathbb{K}[\eta_i])\Big]^T \cdot \Big[t^{c\langle \eta_{i},\kappa_{j}\rangle} \Big]^{-1} \cdot \Big[ P(\mathbb{H}[\kappa_j] )\Big] .\end{equation*}

Equation (3.6) can be written in terms of $R(\mathbb{G};x,y)$, but the resulting expression is not as clean. This is also seen in Negami’s work [Reference Negami21, Corollary 4.7] where his splitting formula is not as clean when expressed in terms of $T(G;x,y)$.

The proof of Theorem 1 is fairly robust, with two key requirements in the argument being that the deletion-contraction recursion relation reduces $P(\mathbb{G})$ to its value on edgeless arrow presentations, and that the value of the polynomial on such arrow presentations is just the number of circles. From Equation (2.3) we see that the topological transition polynomial shares these two properties. Indeed it is straightforward to extend Theorem 1 to the topological transition polynomial.

Theorem 2. Let $\mathbb{G}$ be an arrow presentation that is r-separated into $\mathbb{H}$ and $\mathbb{K}$. The r-separation is defined by 2r welding-arcs. Let $\{\eta_{i}:i=1,\dots d\}$ be an ordering of the set of pairings of $\mathbb{H}$-ends, and $\{\kappa_{j}:j=1,\dots d\}$ be an ordering of the set of pairings of $\mathbb{K}$-ends. Then

\begin{equation*}Q(\mathbb{G}) = \Big[Q(\mathbb{K}[\eta_i])\Big]^T \cdot \Big[t^{c\langle \eta_{i},\kappa_{j}\rangle} \Big]^{-1} \cdot \Big[ Q(\mathbb{H}[\kappa_j] )\Big] ,\end{equation*}

where the square brackets denote matrices or vectors in the obvious way.

We omit the proof of Theorem 2 as it follows the proof of Theorem 1 with only minor modifications. (The deletion-contraction relations from Equation (3.1) that are used in the proof are replaced by the deletion, contraction, and twist-contraction relations of Equation (2.3), and the other small changes follow immediately from this.)

Example 1. Consider the arrow presentation $\mathbb{G}$ consisting of one circle and four arrows with labels in $\{a,b\}$ such that, when travelling round the circle, all arrows point in the same direction and the labels are met in the cyclic order $a,b,a,b$. Thus $\mathbb{G}$ represents the genus 1 ribbon graph consisting of one vertex and two edges (the two edges forming the meridian and longitude of a torus).

The arrow presentation $\mathbb{G}$ has a 2-separation into one-edge arrow presentations $\mathbb{H}$ and $\mathbb{K}$. Name the welding-arcs $1,2,3,4$ so that, together with the arrows, they are met in the order $a,1,b,2,a,3,b,4$ when travelling round the circle. Use these names to order the pairings of the $\mathbb{H}$-ends and $\mathbb{K}$-ends $\{\{1,2\}, \{3,4\}\}$, $\{\{1,3\}, \{2,4\}\}$, $\{\{1,4\}, \{2,3\}\}$. Then we have

\begin{align*} P(\mathbb{G}) &= \Big[P(\mathbb{K}[\eta_i])\Big]^T \cdot \Big[t^{c\langle \eta_{i},\kappa_{j}\rangle} \Big]^{-1} \cdot \Big[ P(\mathbb{H}[\kappa_j] )\Big] \\ &= \begin{bmatrix} xt^2+yt \\ xt+yt \\xt+yt^2\end{bmatrix}^T \cdot \begin{bmatrix} t^2 & t & t \\t& t^2 & t\\ t&t&t^2 \end{bmatrix}^{-1} \cdot \begin{bmatrix} xt+yt^2 \\ xt+yt \\xt^2+yt\end{bmatrix} \\ &= x^2t+2xyt^2+y^2t. \end{align*}

We consider this example in more depth. Here $\eta_2=\kappa_2=\{\{1,3\}, \{2,4\}\}$. When deleting or contracting either edge of $\mathbb{G}$ the arrow presentations $\mathbb{H}[\kappa_2]$ and $\mathbb{K}[\eta_2]$ never arise. Since the proof of Theorem 1 only actually requires us to consider the $\mathbb{H}[\kappa_i]$ and $\mathbb{K}[\eta_i]$ that arise by applying deletion and contraction to all the $\mathbb{H}$-edges or all the $\mathbb{K}$-edges, this means that we can omit η ₂ and κ ₂ from our calculation, giving

(3.13)\begin{equation} \begin{split} P(\mathbb{G}) &= \begin{bmatrix} xt^2+yt \\xt+yt^2\end{bmatrix}^T \cdot \begin{bmatrix} t^2 & t \\t& t^2 \end{bmatrix}^{-1} \cdot \begin{bmatrix} xt+yt^2 \\xt^2+yt\end{bmatrix} \\ &= x^2t+2xyt^2+y^2t. \end{split} \end{equation}

Consequently, if we have some structure on $\mathbb{G}$ that enables us to exclude some pairings then we can reduce the size of the matrices in Theorem 1. (The general result requires us to consider all pairings as we may not know in advance which $\mathbb{H}[\kappa_i]$ and $\mathbb{K}[\eta_i]$ are realized by deletions and contraction.) We make use of these observations in the results that follow.

3.3. The plane case

We shall now consider Theorem 1 in the case where the arrow presentation $\mathbb{G}$ is plane. This situation will reappear in the next section.

Observe that if $\mathbb{G}$ is plane then the arrow presentations $\mathbb{K}[\eta_i]$ and $\mathbb{H}[\kappa_j]$ appearing in Theorem 1 may not be plane. However, for certain applications (for example to deduce an expression for the Jones polynomial of an alternating link, or for the Tutte polynomial) it is required that only plane arrow presentations are considered. We shall now describe how to adapt Theorem 1 so that it holds for the class of plane arrow presentations. In essence this is the situation considered by Burgos in [Reference Burgos6] but removed from the knot theory context.

For example, the set $\{1,2,3,4\}$ has exactly two non-crossing pairings, $\{\{1,2\}, \{3,4\}\}$ and $\{\{1,4\}, \{2,3\}\}$. The set $\{1,2,\ldots, 6\}$ has five non-crossing pairings, $\{\{1,2\} ,\{3,4\},\{5,6\} \}$, $\{\{1,4\} ,\{2,3\},\{5,6\} \}$, $\{\{1,6\} ,\{2,3\},\{4,5\} \}$, $\{\{1,6\} ,\{2,5\},\{3,4\} \}$, $\{\{1,2\} ,\{3,6\},\{4,5\} \}$.

Now suppose that $\mathbb{G}$ is a plane arrow presentation that is r-separated into $\mathbb{H}$ and $\mathbb{K}$. For simplicity, throughout this subsection, we assume that $\mathbb{H}$ and $\mathbb{K}$ represent connected ribbon graphs. We describe a process for drawing the arrow presentation $\mathbb{G}$ on the sphere. Since $\mathbb{G}$ is a plane arrow presentation it describes a plane ribbon graph, which, by the classification of surfaces with boundary, has a unique embedding in the sphere (up to homeomorphism of the sphere). This embedding necessarily induces a drawing of its original arrow presentation on the sphere.

As $\mathbb{H}$ and $\mathbb{K}$ define an r-separation, there is a simple closed curve on this sphere that intersects the drawing of $\mathbb{G}$ only in the welding-arcs and encloses all of the arrows of $\mathbb{H}$ and all of the circles of $\mathbb{H}$ that are not also circles of $\mathbb{K}$. We call this curve the separating curve. By choosing a direction and a starting point, and then following the separating curve, we obtain a linear order on the set of welding-arcs, and hence also on their $\mathbb{K}$-ends and $\mathbb{H}$-ends. We call such orders plane orders. Thus we may consider non-crossing pairings of the $\mathbb{K}$-ends and $\mathbb{H}$-ends with respect to a plane order.

For example, Figure 4a) shows a plane arrow presentation drawn on the plane, with a 2-separation and its separating curve indicated by the dotted line.

Note that in the case where the 2r welding-arcs lie on exactly r circles, it is not hard to define a plane order intrinsically to the original arrow presentation $\mathbb{G}$ (i.e., without reference to a plane embedding). However, we do not know how to do this in general.

Figure 4. Drawings of arrow presentations with welding-arcs given a plane order. (a) A plane arrow presentation. (b) A partial dual of a plane arrow presentation. This is non-plane.

We want to adapt Theorem 1 and its proof to the case where $\mathbb{G}$ is plane. For this we follow the approach indicated in the observation at the end of Example 1. When $\mathbb{G}$ is plane, by Proposition 3 only non-crossing pairings with respect to a plane order appear in Equations (3.9), (3.10), and (3.12), and therefore only these pairings need be considered in Equation (3.6). In this case, by Proposition 2 only plane arrow presentations appear in Equation (3.6). So we have the following.

Theorem 3. Let $\mathbb{G}$ be a plane arrow presentation that is r-separated into $\mathbb{H}$ and $\mathbb{K}$, and consider a plane order of the welding-arcs. With respect to this order let $\{\eta_{i}:i=1,\dots d\}$ be the set of non-crossing pairings of $\mathbb{H}$-ends, and $\{\kappa_{j}:j=1,\dots d\}$ be the set of non-crossing pairings of $\mathbb{K}$-ends. Then

(3.14)\begin{equation} P(\mathbb{G}) = \Big[P(\mathbb{K}[\eta_i])\Big]^T \cdot \Big[t^{c\langle \eta_{i},\kappa_{j}\rangle} \Big]^{-1} \cdot \Big[ P(\mathbb{H}[\kappa_j] )\Big] ,\end{equation}

where all the arrow presentations are plane.

As $\mathbb{G}$ is plane, Theorem 3 could be written in terms of the Tutte polynomial $T(G;x,y)$ or Negami’s polynomial $f(G;t,x,y)$. However, this re-writing would not result in Negami’s splitting formula since the graph and ribbon graph polynomials act differently on loops.

4.1. Partial duals of plane graphs

Suppose an arrow presentation $\mathbb{G}$ has a plane partial dual, and that $\mathbb{G}$ is r-separated into $\mathbb{H}$ and $\mathbb{K}$. (Note that $\mathbb{G}$ need not be plane. For example, the arrow presentation in Figure 4b) is non-plane but the plane arrow presentation in Figure 4a) is one of its partial duals.) Let A be any set of its edges with the property that $\mathbb{G}^{A}$ is plane. (A suitable set A exists by definition. Moreover it is straightforward to find such a set A. They are those that define “plane-biseparations” as described in [Reference Moffatt19, Reference Moffatt20].) Observe that the arrow presentations induced from $\mathbb{H}$ and $\mathbb{K}$ under the same partial duality with respect to A are also plane.

Since partial duality operates locally at the pairs of arrows with the same label in the arrow presentation (as is seen in Figure 1), it follows that $\mathbb{G}^A$ is r-separated into $\mathbb{H}^{A\cap E(\mathbb{H})}$ and $\mathbb{K}^{A\cap E(\mathbb{K})}$, and this r-separation uses the same welding-arcs.

We proceed by drawing the arrow presentation $\mathbb{G}$ on the sphere. To do this, we follow the process in $\S$ 3.3 to draw the plane arrow presentation $\mathbb{G}^{A}$ on the sphere. We also draw a separating curve, again just as in $\S$ 3.3, and then modify the drawing of $\mathbb{G}^{A}$ to one of $\mathbb{G}$ by forming the partial dual $(\mathbb{G}^{A})^A =\mathbb{G}$. We do this by splicing the drawing following Figure 1 and the definition of a partial dual. This gives a drawing of the arrow presentation $\mathbb{G}$ on the plane, and by Proposition 1 the welding-arcs for $\mathbb{G}^A$ induce an r-separation for $\mathbb{G}$, and the separating curve for $\mathbb{G}^A$ is also one for $\mathbb{G}$. Read off the plane order of the welding-arcs for $\mathbb{G}$ from this drawing. We say that any order of the welding-arcs of $\mathbb{G}$ obtained in this way is an induced plane order. Note that it does depend upon the choice of A. Figure 4a) shows a plane arrow presentation $\mathbb{G}^{\{b,c,e\}}$, while Figure 4b) shows a drawing of $\mathbb{G}$ constructed as above.

We can collect this together to adapt Theorem 1 as follows. When $\mathbb{G}$ has a plane partial dual, by Proposition 5 it is enough to consider only the non-crossing pairings with respect to an induced plane order in the proof of Theorem 1. Also, by Proposition 4 only arrow presentations that have plane partial duals appear in Equation (3.6). So we have the following.

Theorem 4. Let $\mathbb{G}$ be an arrow presentation that has a plane partial dual and that is r-separated into $\mathbb{H}$ and $\mathbb{K}$, and consider an induced plane order of the welding-arcs. With respect to this order let $\{\eta_{i}:i=1,\dots d\}$ be the set of non-crossing pairings of $\mathbb{H}$-ends, and $\{\kappa_{j}:j=1,\dots d\}$ be the set of non-crossing pairings of $\mathbb{K}$-ends. Then

(4.1)\begin{equation} P(\mathbb{G}) = \Big[P(\mathbb{K}[\eta_i])\Big]^T \cdot \Big[t^{c\langle \eta_{i},\kappa_{j}\rangle} \Big]^{-1} \cdot \Big[ P(\mathbb{H}[\kappa_j] )\Big] ,\end{equation}

where all the arrow presentations have plane partial duals.

4.2. A splitting formula for the Kauffman bracket

For this subsection we assume a familiarity with basic knot theory, and in particular with the Kauffman bracket and Jones polynomial.

There is a classical and well-known way of representing an alternating link diagram as a plane graph, called a Tait graph (see for example the books [Reference Brylawski and Oxley5, Reference Ellis-Monaghan and Moffatt12, Reference Welsh27]). Through this the Jones polynomial of an alternating link can be expressed as an evaluation of the Tutte polynomial of a plane graph. This connection was central to Thistlethwaite’s proof of the Tait conjectures [Reference Thistlethwaite26] and Jaeger, Vertigan, and Welsh’s work on the computational complexity of the Jones polynomial in [Reference Jaeger, Vertigan and Welsh18].

If instead the link diagram is non-alternating then it can be encoded in a non-plane ribbon graph, or equivalently a non-plane arrow presentation, and the Jones polynomial can be recovered as an evaluation of the ribbon graph polynomial. This result, shown in [Reference Dasbach, Futer, Kalfagianni, Lin and Stoltzfus10], is a direct extension of the classical relationship between the Jones and Tutte polynomials.

Let $D\subset \mathbb{R}^2$ be a link diagram with labelled crossings. An A-smoothing of a crossing of D is the replacement of the crossing with a pair of curves as indicated in Figures 5a) and 5b). An arrow-marked A-smoothing of a crossing c of D, is the replacement of the crossing c with a pair of decorated curves as indicated in Figures 5a) and 5c). The pair of arrows in the figure are given the same label as the crossing. If every crossing of D is replaced with an arrow-marked A-smoothing, then the resulting diagram is an arrow presentation. This arrow presentation describes a ribbon graph referred to as the all-A ribbon graph of D as introduced in [Reference Dasbach, Futer, Kalfagianni, Lin and Stoltzfus10]. We use $\mathbb{A}(D)$ to denote its arrow presentation. An example is shown in Figure 5.

Figure 5. A ribbon graph of a link diagram. (a) A crossing c. (b) An A-smoothing of c. (c) An arrow-marked A-smoothing of c. (d) A (non-alternating) link diagram D. (e) The (non-plane) arrow presentation $\mathbb{A}(D)$.

Not every ribbon graph arises as the all-A ribbon graph of a link diagram. But we have the following characterization.

A proof can be found in [Reference Moffatt19], but it is also implicit in [Reference Chmutov8].

We now describe a splitting of link diagrams which corresponds to r-separation.

Let T_K and T_H be 2r-tangle diagrams with the open ends of each tangle numbered from 1 to 2r as in Figure 6a). Let D be the link diagram obtained by identifying corresponding open ends of the two tangles, as in Figure 6b).

Figure 6. Two tangles and a resulting link diagram.

If η is a non-crossing pairing of $\{1,\ldots, 2r\}$ then we define $T_K[\eta]$ to be the link diagram obtained by closing the tangle T_K by η. (So non-intersecting arcs are drawn to connect the open ends of T_K that are paired in η.) Two non-crossing pairings are of particular interest. A-smoothing each crossing of T_K induces a non-crossing pairing $\alpha(T_K)$ of the ends of T_K (or equivalently, the numbers $1,\ldots ,2r$). Let $\widehat{T}_H$ denote the link diagram $T_H[\alpha(T_K)]$. Define $\widehat{T}_K$ analogously.

It is clear that the arrow presentation $\mathbb{A}(D)$ is r-separated into $\mathbb{A}(\widehat{T}_H)$ and $\mathbb{A}(\widehat{T}_K)$. Moreover, the welding-arcs correspond to the ends of the tangles T_K and T_H with the numbering of these tangle ends giving a plane ordering of the welding-arcs. Furthermore, the operation of constructing a ribbon graph by A-smoothing a link diagram commutes with the operation of closing ends with a non-crossing pairing, so

(4.2)\begin{equation} \mathbb{A}(\widehat{T}_K)[\eta] = \mathbb{A}({T}_K[\eta]). \end{equation}

The Kauffman bracket $\langle D \rangle$ (and hence the Jones polynomial) of any link diagram D can be obtained from $P(\mathbb{A}(D))$:

(4.3)\begin{equation} \langle D \rangle = (-A^2-A^{-2})^{-1}\; P(\mathbb{A}(D); \;A^{-1},\; A,\; -A^2-A^{-2}). \end{equation}

This follows from a relationship between $R(\mathbb{A}(D);x,y)$ and the Kauffman bracket from [Reference Dasbach, Futer, Kalfagianni, Lin and Stoltzfus10] together with Equation (3.2). (Alternatively, it is not hard to deduce the identity directly from the definitions of the polynomials.) Now, from Theorem 4, we can recover Burgos’ splitting formula for the Jones polynomial [Reference Burgos6].

Theorem 5. Let T_H and T_K be 2r-tangle diagrams with the open ends of each tangle numbered 1 through 2r as in Figure 6. Let D be the link diagram obtained by identifying corresponding open ends of the two tangles.

With respect to this order let $\{\eta_{i}:i=1,\dots d\}$ be the set of non-crossing pairings of the open ends of T_H, and $\{\kappa_{j}:j=1,\dots d\}$ be the set of non-crossing pairings of the open ends of T_K. Then

(4.4)\begin{equation} \langle D \rangle = \Big[ \langle T_K[\eta_i] \rangle \Big]^T \cdot \Big[(-A^2-A^{-2})^{c\langle \eta_{i},\kappa_{j}\rangle-1} \Big]^{-1} \cdot \Big[ \langle T_H[\kappa_j] \rangle \Big]. \end{equation}

Proof. To simplify the notation we write $P(\mathbb{G})$ for $P(\mathbb{G}; \;A^{-1}, \;A, \;-A^2-A^{-2})$, and t for $-A^2-A^{-2}$. With this we have

\begin{align*} t \cdot \langle D \rangle &= P(\mathbb{A}(D)) \\ &= \Big[ P(\mathbb{A}(\widehat{T}_K)[\eta_i]) \Big]^T \cdot \Big[t^{c\langle \eta_{i},\kappa_{j}\rangle} \Big]^{-1} \cdot \Big[ P( \mathbb{A}(\widehat{T}_H )[\kappa_j]) \Big] \\ &= \Big[ P(\mathbb{A}({T}_K[\eta_i])) \Big]^T \cdot \Big[t^{c\langle \eta_{i},\kappa_{j}\rangle} \Big]^{-1} \cdot \Big[ P( \mathbb{A}({T}_H [\kappa_j] ))\Big] \\ &= \Big[ t\cdot \langle{T}_K[\eta_i]\rangle \Big]^T \cdot \Big[t^{c\langle \eta_{i},\kappa_{j}\rangle} \Big]^{-1} \cdot \Big[t\cdot \langle {T}_H [\kappa_j] \rangle \Big] \\ &= t\cdot \Big[ \langle{T}_K[\eta_i]\rangle \Big]^T \cdot \Big[t^{c\langle \eta_{i},\kappa_{j}\rangle-1} \Big]^{-1} \cdot \Big[ \langle {T}_H [\kappa_j] \rangle \Big]. \end{align*}

Here the first equality is from (4.3), the second is Theorem 4, the third is from (4.2), and the fourth is from (4.3).