1 Introduction↩︎

1.1 Torsor structures on spanning trees of plane graphs↩︎

Given a connected finite graph \(G\), there is — in general — no natural way to make the set \(\mathcal{T}(G)\) of spanning trees of \(G\) into a group.

For example, if \(G=C_3\) is a 3-cycle, then there is clearly no canonical choice for an “identity tree”. On the other hand, if \(C_3\) is embedded in the plane as an equilateral triangle and we orient the plane counterclockwise, then there is a natural simply transitive action of the group \(\mathbb{Z}/ 3\mathbb{Z}\) on \(\mathcal{T}(C_3)\) given by counterclockwise rotation. Moreover, the group \(\mathbb{Z}/3\mathbb{Z}\) is isomorphic to the Jacobian group¹ of \(C_3\).

For any connected graph \(G\), the cardinality of its Jacobian group \(\mathrm{Jac}(G)\) is equal to the number of spanning trees of \(G\) (this is a version of Kirchhoff’s Matrix-Tree Theorem). So a better question is whether there is a natural way to define a simply transitive action of \(\mathrm{Jac}(G)\) on \(\mathcal{T}(G)\).

In general, the answer seems to be no (depending on precisely what one means by “natural”). However, for graphs embedded in the plane, the answer is yes (at least once we fix an orientation of the plane). There are at least three different ways to describe such an action:

1. In [1], Chan, Church, and Grochow construct, for any connected graph \(G\), a family of simply transitive actions of \(\mathrm{Jac}(G)\) on \(\mathcal{T}(G)\) using the rotor-routing process, and they show that for plane graphs the resulting action is independent of all choices (up to isomorphism).

2. In [2], the first author and Wang define a different family of simply transitive actions of \(\mathrm{Jac}(G)\) on \(\mathcal{T}(G)\) using the Bernardi process and show that, for plane graphs, the resulting action is independent of all choices and coincides with the action defined in [1]. We refer collectively to these actions as “Bernardi torsors”.²

3. In [3], Backman, the first author, and Yuen define the Jacobian of a regular matroid \(M\) and construct a family of simply transitive actions of \(\mathrm{Jac}(M)\) on the set \(\mathcal{B}(M)\) of bases of \(M\), which generalizes Yuen’s work [4]. We refer collectively to these actions as “BBY torsors”. When \(M = M(G)\) is the cycle matroid associated to a plane graph \(G\), using [4], the authors of [3] show that there is a canonical choice for the action of \(\mathrm{Jac}(M(G))\) on \(\mathcal{B}(M(G)) = \mathcal{T}(G)\), and that this action coincides with the one defined in [1] and [2], as described above.

1.2 An extension to the non-planar case↩︎

For a while after the publication of [1]–[3], it seemed that the considerations in §1.1 might be the end of the story: for a plane graph \(G\) there is a natural way to define a simply transitive group action on \(\mathcal{T}(G)\), and for a non-planar graph there is not. However, when we came across the paper [5] by Merino, Moffatt, and Noble, we realized that it should be possible to generalize the results of [1]–[3] to the non-planar case from their new perspective.

The key idea is to consider connected graphs \(G\) embedded³ on an arbitrary compact and connected oriented surface \(\Sigma\), replacing the Jacobian group of \(G\) with a group that depends on the embedding and the set of spanning trees of \(G\) with a larger set that also depends on the embedding. We refer to such a pair \(\mathbf{G}= (G,\Sigma)\) as a ribbon graph⁴. The reason for the name “ribbon graph” is that one often works instead with an \(\epsilon\)-thickening of the graph \(G\) in \(\Sigma\) for a suitably small \(\epsilon > 0\), which is a 2-dimensional manifold-with-boundary that resembles a ribbon. We use the two viewpoints interchangeably in this paper.

When \(\Sigma\) is a sphere, the ribbon graph \(\mathbf{G}\) encodes the same data as \(G\) together with a plane embedding and an orientation of the plane. However, every graph \(G\) can be realized as a ribbon graph \(\mathbf{G}= (G,\Sigma)\) for a suitable surface \(\Sigma\).

For a ribbon graph \(\mathbf{G}\), one defines a spanning quasi-tree to be a connected subgraph \(Q\) of \(G\) which contains every vertex of \(G\), and such that an \(\epsilon\)-thickening of \(Q\) in \(\Sigma\) has exactly one boundary component; see Example 2. When \(\Sigma\) is a sphere, this is the same thing as a spanning tree of \(G\), but for surfaces of higher genus the two notions diverge; in general, every spanning tree is also a spanning quasi-tree, but not vice-versa.

The main result of this paper can be stated as follows:

1.3 The Jacobian of a ribbon graph and regular orthogonal matroids↩︎

The group \(\mathrm{Jac}(\mathbf{G})\) acting simply and transitively on \(\mathcal{Q}(\mathbf{G})\) in Theorem 1 is isomorphic to the critical group of \(\mathbf{G}\) as defined by Merino, Moffatt, and Noble [5]. We call it the Jacobian group of the ribbon graph \(\mathbf{G}\). Among other things, the authors prove in [5] that the cardinality of \(\mathrm{Jac}(\mathbf{G})\) is the number of spanning quasi-trees in \(\mathbf{G}\), and this is one of the key observations which motivated the present paper.

In [6], [7], Bouchet showed that there is a natural way to associate to a ribbon graph \(\mathbf{G}\) an orthogonal matroid⁵ \(M(\mathbf{G})\). Orthogonal matroids are natural generalizations of matroids, and in particular there is a special class of orthogonal matroids, called regular orthogonal matroids, which generalize regular matroids. Roughly speaking, a regular orthogonal matroid \(M\) is one which is representable by a principally unimodular skew-symmetric matrix, and we call the choice of such a matrix a representation of \(M\). The work of Bouchet shows that not only is the orthogonal matroid \(M(\mathbf{G})\) associated to a ribbon graph \(\mathbf{G}\) regular, it also comes equipped with a natural equivalence class of regular representations.

We generalize the results of [5] by showing that there is a natural way to associate a Jacobian group \(\mathrm{Jac}(M, [\mathcal{C}])\) to a pair \((M,[\mathcal{C}])\) consisting of a regular orthogonal matroid \(M\) and an equivalence class \([\mathcal{C}]\) of regular representations of \(M\). The cardinality of \(\mathrm{Jac}(M,[\mathcal{C}])\) is equal to the number of bases of \(M\). Moreover, when \(\mathcal{C}\) arises from a ribbon graph \(\mathbf{G}\) via the construction of Bouchet, \(\mathrm{Jac}(M,[\mathcal{C}])\) coincides with the critical group of Merino et al.⁶

As we now explain, our construction of a simply transitive action of \(\mathrm{Jac}(\mathbf{G})\) on \(\mathcal{Q}(\mathbf{G})\) for a ribbon graph \(\mathbf{G}\) is a special case of a more general family of simply transitive actions of \(\mathrm{Jac}(M,[\mathcal{C}])\) on the set of bases of \(M\).

1.4 Generalized BBY torsors for regular orthogonal matroids↩︎

In order to explain our generalization of the theory of “BBY torsors” from regular matroids to regular orthogonal matroids, we first briefly recall the setup from Backman et al. [3]. We assume in this discussion that the reader is familiar with the basic terminology of matroid theory.

If \(M\) is a regular matroid on the ground set \(E\) then \(M\) is uniquely orientable (up to a certain natural notion of equivalence of orientations). In particular, each circuit \(C\) of \(M\) has two distinguished orientations \(\pm \vec{C}\). A choice of one of these two orientations, call it \(\sigma(C)\), for each circuit \(C\) of \(M\) is called a circuit signature for \(M\). Such a signature is called acyclic if \(\sum_C a_C \sigma(C) = 0\) in \(\mathbb{R}^E\) with all \(a_C \geq 0\) implies \(a_C=0\) for all circuits \(C\). Applying the same definitions to the dual matroid yields the notion of (acyclic) cocircuit signature.

If \(\mathcal{O},\mathcal{O}'\) are orientations of \(E\) (meaning we choose one of two possible states for each element of \(E\)), we call \(\mathcal{O}\) and \(\mathcal{O}'\) equivalent, and write \(\mathcal{O}\sim \mathcal{O}'\), if we can get from \(\mathcal{O}\) to \(\mathcal{O}'\) by a sequence of circuit and cocircuit reversals (meaning that whenever we have an orientation consistent with a signed circuit or cocircuit \(\vec{C}\), we can replace \(\vec{C}\) in the orientation by \(-\vec{C}\)). The set \(\mathcal{R}(M)\) of equivalence classes of orientations of \(E\) is called the circuit-cocircuit reversal system for \(M\).⁷

The authors of [3] prove:

If \(G\) is a (connected) plane graph and \(M=M(G)\), then \(\mathcal{B}(M) = \mathcal{T}(G)\) and circuits of \(M\) coincide with (simple) cycles of \(G\), and there is a canonical choice for \(\sigma\) given by orienting all cycles of \(G\) counterclockwise. Applying the same reasoning to a plane dual of \(G\) allows us to similarly define an acyclic cocircuit signature \(\sigma^*\). It turns out that different plane duals give rise to the same group actions, from which one deduces that there is in fact a canonical simply transitive action of \(\mathrm{Jac}(G)\) on \(\mathcal{T}(G)\). This is the same action discussed in §1.1 above, which had previously been constructed in different ways in [1] and [2].

We apply similar ideas in this paper to the case of regular orthogonal matroids. In order to explain, we first recall from [11] that an orthogonal matroid \(M\) on the ground set \(E \cup E^* = \{ 1,\ldots,n,1^*,\ldots,n^* \}\) can be characterized either in terms of its bases, which are certain “transversal” subsets of \(E \cup E^*\) not containing any skew pairs \(\{ i,i^* \}\), or in terms of its circuits, which are also subsets of \(E \cup E^*\). Given a matroid \(N\) on \(E\), there is an associated orthogonal matroid on \(E \cup E^*\), called the lift of \(N\), whose set of bases is \(\{ B \cup (E-B)^* \; | \; B \in \mathcal{B}(N) \}\) and whose circuits are either circuits of \(N\) or stars of cocircuits of \(N\) (i.e., \(\{ i^* \; | \; i \in D\}\) for some cocircuit \(D\) of \(N\)). In particular, one can think of the circuit set of an orthogonal matroid as generalizing both circuits and cocircuits of a matroid. A matroid is regular if and only if its corresponding orthogonal matroid is regular in the sense of [11], which coincides with the underlying even delta-matroid being regular in the sense of [12].

Given a regular orthogonal matroid \(M\) on the ground set \(E \cup E^*\) and a regular representation \(\mathcal{C}\) of \(M\), we will define an equivalence relation on the set of orientations of \(E\) through a suitable notion of circuit reversals. We will then define a corresponding circuit reversal system \(\mathcal{R}(M,\mathcal{C})\), cf. Definition 10. When \(M\) arises from a regular matroid \(\tilde{M}\), \(\mathcal{R}(M,\mathcal{C})\) coincides with the circuit-cocircuit reversal system of \(\tilde{M}\) as defined in [10].

We will also define a notion of acyclic circuit signature for \((M,\mathcal{C})\), which also generalizes the corresponding notion from [3]. With these concepts in hand, we have:

If \(\mathbf{G}= (G,\Sigma)\) is a ribbon graph and \((M,\mathcal{C})\) is the corresponding pair consisting of a regular orthogonal matroid \(M\) and a regular representation \(\mathcal{C}\) then \(\mathcal{B}(M) = \mathcal{Q}(\mathbf{G})\), and choosing a generic point \(p\) of the surface \(\Sigma\) allows us to define an acyclic circuit signature \(\sigma_p\) using the orientation of \(\Sigma\). We show that different choices of \(p\) lead to isomorphic simply transitive group actions, and then Theorem 1 follows from Theorem 2.

1.5 A few words on the proof of Theorem 2↩︎

Although the statement of Theorem 2 is similar to the analogous statement in [3] (in the case of regular matroids), there are some complications which arise in the orthogonal case.

First of all, the proof in [3] that \(\bar{\beta} : \mathcal{B}(M) \to \mathcal{R}(M)\) is a bijection involves geometric reasoning related to tilings of a certain zonotope. It is not clear how to define the corresponding geometric objects in the more general setting of regular orthogonal matroids, so we employ the machinery introduced in the second author’s paper [13], in which a purely combinatorial proof of the bijectivity of \(\bar{\beta}\) is given. This involves, among other things, a discussion of fourientations in the sense of [14] and an analogue for orthogonal matroids of Farkas’s Lemma from oriented matroid theory.

Second, we need to prove that the definition we give for \(\mathrm{Jac}(M,\mathcal{C})\) coincides with the definition in [5] when \((M,\mathcal{C})\) comes from a ribbon graph \(\mathbf{G}\).

Third, the descriptions in the existing literature of the circuits of the orthogonal matroid associated to a ribbon graph \(\mathbf{G}\) are hard to work with in the present context, since they are not very geometric in nature. We provide an alternate description which should be of independent interest. We must also connect this to the oriented circuits of a regular orthogonal matroid and relate our description to the regular representation defined by Bouchet.

1.6 Bernardi torsors for ribbon graphs↩︎

As mentioned above, BBY torsors generalize Bernardi torsors in the case of graphs. We show that a similar phenomenon holds in the context of ribbon graphs for our generalized BBY torsors.

The idea is as follows (see §5.3 for details). Let \(\mathbf{G}= (G,\Sigma)\) be a ribbon graph. Let \((M,\mathcal{C})\) be the corresponding pair consisting of an orthogonal matroid \(M\) and an orthogonal representation \(\mathcal{C}\) of \(M\), as defined by Bouchet. Let \(\mathbf{G}^* = (G^*,\Sigma)\) be a dual ribbon graph embedded in \(\Sigma\), whose edges we call coedges of \(G\).

Let \(Q\) be a spanning quasi-tree of \(\mathbf{G}\), and thicken \(Q\) to a “ribbon” \(N\). Starting at an arbitrary point of the boundary \(\partial N\) of \(N\), travel along \(\partial N\) according to the orientation of \(\Sigma\) (as seen from the interior of \(N\)). For any edge \(e\), you will either traverse \(e\) twice, if \(e\notin Q\), or traverse \(e^*\) twice, if \(e\in Q\). The first time you traverse \(e\), orient \(e\) inward toward \(N\), and the first time you traverse \(e^*\), orient \(e\) in the direction you’re traveling. Since \(Q\) has a unique boundary component, you will eventually get back to your starting point, and when you do, you will have oriented each edge \(e\) of \(G\), cf. Figure 7. This defines an orientation \(\mathcal{O}\) on the edges of \(G\), which has a corresponding circuit-reversal class \([\mathcal{O}]\) in \(\mathcal{R}(M,\mathcal{C})\).

We show that this procedure gives a bijection from \(\mathcal{Q}(\mathbf{G})\) to \(\mathcal{R}(M,\mathcal{C})\). Moreover, while different choices of starting point along \(\partial N\) will in general give different bijections, it follows from our results that if \(\beta_1,\beta_2\) are any two such bijections, there exists \(g \in \mathrm{Jac}(M,\mathcal{C})\) such that \(\beta_2(Q) = g \cdot \beta_1(Q)\) for all \(Q \in \mathcal{Q}\). Our results also show that the resulting “Bernardi torsor” coincides with the one defined in [2] when \(\Sigma\) is a sphere (i.e., when \(\mathbf{G}\) is a plane ribbon graph). In addition (and this is how our proof actually proceeds), we show that this torsor is in fact a BBY torsor with respect to a suitable acyclic circuit signature \(\sigma\) of \((M,\mathcal{C})\).

1.7 Structure of the paper↩︎

In §2, we provide background information on orthogonal matroids, oriented and regular orthogonal matroids, and ribbon graphs, and we prove the above-mentioned generalization of Farkas’s Lemma.

In §3, we define the Jacobian group \(\mathrm{Jac}(\mathcal{C})\) of a pair \((M,\mathcal{C})\), where \(M\) is a regular orthogonal matroid and \(\mathcal{C}\) is a regular representation. We also define the circuit reversal system \(\mathcal{R}(\mathcal{C})\) in this context, and prove that there is a canonical group action of \(\mathrm{Jac}(\mathcal{C})\) on the set \(\mathcal{R}(\mathcal{C})\). Lastly, we prove that the group action is invariant under reorientations of \(\mathcal{C}\).

In §4, we formulate and prove Theorem 2, generalizing the theory of BBY bijections from regular matroids to regular orthogonal matroids.

In §5, we apply Theorem 2 to the case of ribbon graphs and deduce Theorem 1. We also relate our generalized BBY bijections to the family of Bernardi torsors associated to a ribbon graph \(\mathbf{G}\).

In §6, we examine how Theorem 1 interacts with dual ribbon graphs and explain how our definitions and results generalize the corresponding ones for regular matroids.

In Appendix 7, we discuss the compatibility between certain definitions in the existing literature on ribbon graphs and the definitions employed in the main body of this paper.

2 Preliminaries↩︎

2.1 Orthogonal matroids↩︎

Let \(E\) be a finite set of cardinality \(n\) and let \(E^* := \{e^* : e\in E\}\) be a disjoint copy of \(E\). For \(e\in E\), we write \((e^*)^* = e\).

A skew pair is a \(2\)-element subset \(\{e,e^*\}\). A transversal is an \(n\)-element subset of \(E\cup E^*\) which does not contain any skew pair. A subtransversal⁸ is a subset of a transversal.

Recall that a matroid is a pair \((E,\mathcal{B})\) such that \(\mathcal{B}\) is a nonempty collection of subsets of \(E\) satisfying:

• For all \(B,B'\in \mathcal{B}\) and \(e\in B\setminus B'\), there is an element \(f\in B'\setminus B\) such that \((B\setminus \{e\}) \cup \{f\} \in \mathcal{B}\).

Orthogonal matroids generalize matroids: given a matroid \(N = (E,\mathcal{B})\), the pair \(\mathrm{lift}(N):=(E\cup E^*, \mathcal{B}')\) is an orthogonal matroid, where \(\mathcal{B}' = \{B\cup (E\setminus B)^* : B\in \mathcal{B}\}\). The set of circuits of \(\mathrm{lift}(N)\) is the union of \(\{C\subseteq E: \text{C is a circuit of N}\}\) and \(\{D^*\subseteq E^*: \text{D is a cocircuit of N}\}\).

Orthogonal matroids have a “cryptomorphic” characterization in terms of circuits which generalizes both the circuit elimination axiom for matroids and the fact that circuits and cocircuits of a matroid are orthogonal. The following theorem is due to Bouchet [15], [16] and is a special case of a more general characterization of multimatroids in terms of their circuits. In particular, orthogonal matroids coincide with tight 2-matroids in [15], [16]. For convenience, we refer the reader to [17].

The fundamental circuit of an orthogonal matroid \(M\) with respect to a basis \(B\) and an element \(e\in B^*\) is a circuit contained in \(B\triangle \{e,e^*\}\), which exists and is unique by [item:c3] and [item:c5]. We denote it by \(\mathrm{FC}(B,e)\).

We will need the following lemma later, which strengthens [item:c5].

2.2 Oriented and regular orthogonal matroids↩︎

Wenzel [19], [20] introduced oriented orthogonal matroids as a generalization of oriented matroids. Wenzel defined this concept by generalizing the chirotope axiom for oriented matroids, and later Jin and the third author [11] presented an equivalent definition in terms of signed circuits.

For a vector \(\mathbf{C}\in \mathbb{Z}^{E\cup E^*}\), let \(\mathbf{C}^*\) be the vector in \(\mathbb{Z}^{E\cup E^*}\) such that \(\mathbf{C}^*(e) = \mathbf{C}(e^*)\) for all \(e\in E\cup E^*\).

By [11], an oriented matroid is the same thing as an oriented orthogonal matroid whose underlying orthogonal matroid is equal to \(\mathrm{lift}(N)\) for some matroid \(N\).

Each regular orthogonal representation induces a principally unimodular skew-symmetric matrix as follows. A square matrix is principally unimodular (in short, PU) if it is defined over the rational numbers \(\mathbb{Q}\) and all its principal minors are \(0\) or \(\pm 1\). Note that any principal minor of a PU skew-symmetric matrix is either \(0\) or \(1\), and a skew-symmetric matrix is PU if and only if every principal submatrix has Pfaffian \(0\) or \(\pm 1\). These facts follow from Cayley’s theorem that \(\mathrm{pf}(A)^2 = \det(A)\) for any skew-symmetric matrix \(A\).

Given a regular orthogonal representation \(\mathcal{C}\) of an orthogonal matroid \(M\), we fix a basis \(B\) of \(M\) and let \(\Lambda(\mathcal{C},B)\) be the \(B \times (E\cup E^*)\) matrix defined by taking its \(e\)-th row as \(\mathbf{C}_e \in \mathcal{C}\) such that \(\mathrm{supp}(\mathbf{C}_e) = \mathrm{FC}(B,e^*)\) and \(\mathbf{C}_e(e^*) = 1\). By rearranging the columns, we have \[\begin{align} \label{eq:A-matrix} \Lambda(\mathcal{C},B) = \begin{blockarray}{ccc} \scalebox{0.7}{\text{B^*}} & \scalebox{0.7}{\text{B}} & \\ \begin{block}{(cc)c} I & A & \scalebox{0.7}{\text{B}} \\ \end{block} \end{blockarray} \end{align}\tag{1}\] where \(I\) is the identity matrix. Then \(A\) is skew-symmetric by [item:ror]. On the other hand, the regular orthogonal representation \(\mathcal{C}\) induces a Wick function over the regular partial field \(\mathbb{U}_0\) by [11]; that is, a function \(\varphi\) from the set of transversals to \(\{0,\pm 1\}\) that satisfies the Wick equations; see [11] for details. This function is equivalent, up to normalization so that \(\varphi(B^*) = 1\), to the Pfaffian map introduced by Wenzel [19]. Therefore, by Wenzel’s result [19], the Wick function \(\varphi\) encodes precisely the Pfaffians of \(A\), which implies that \(A\) is PU.

The \(\mathbb{Q}\)-span of \(\mathcal{C}\) has dimension at most \(n\), since it is an isotropic subspace of the \(2n\)-dimensional space \(\mathbb{Q}^{E\cup E^*}\) with respect to the non-degenerate bilinear form \(\langle \cdot, \cdot \rangle\) in [item:ror]; see [21]. The row span of \(\Lambda(\mathcal{C},B)\) over \(\mathbb{Q}\) is equal to the \(\mathbb{Q}\)-span of \(\mathcal{C}\), as the \(n\) rows \(\mathbf{C}_e\)’s of \(\Lambda(\mathcal{C},B)\) are linearly independent. In fact, the same statement holds for \(\mathbb{Z}\)-spans:

The following is a consequence of (4.2) and (4.4) in [22].

As the determinant of nonsingular \(A[B \cap T^*, B \cap T^*]\) equals \(1\), we deduce the following corollary. This result generalizes the Matrix–Quasi-tree Theorem for ribbon graphs [5]; the connection between orthogonal matroids and ribbon graphs is explained in §2.4.

We remark that the matrix \(I+A\) has entries in \(\{0,\pm 1\}\) because the supports of the rows of \(\begin{pmatrix} I & A \end{pmatrix}\) are subtransversals.

Given an oriented orthogonal matroid, there is a way to obtain other oriented orthogonal matroids with the same underlying orthogonal matroid.

A reorientation \(\mathcal{C}^X\) is an oriented orthogonal matroid. Furthermore, if \(\mathcal{C}\) is a regular orthogonal representation, then \(\mathcal{C}^{X}\) is also a regular orthogonal representation.

We briefly review the definition of regular matroids and their connection to regular orthogonal matroids. Regular matroids are usually defined in terms of totally unimodular matrices, but here we recall an equivalent definition in terms of chain group representations in the sense of Tutte [23] or weak \(\mathbb{U}_0\)-circuit sets in the sense of [24].

2.3 Fourientations and a generalized Farkas’ Lemma↩︎

In this section we formulate an extension of Farkas’ Lemma for oriented orthogonal matroids. For a vector \(\mathbf{C}\), we write \(\mathbf{C}\ge 0\) if all entries of \(\mathbf{C}\) are nonnegative.

Farkas’ Lemma for oriented matroids [25] is equivalent to the special case of Lemma 4 where the underlying orthogonal matroid of \(\mathcal{C}\) in Lemma 4 is equal to \(\mathrm{lift}(N)\) for some (oriented) matroid \(N\). We will in fact prove a more general version of Lemma 4, which implies the “\(4\)-painting lemma” for oriented matroids [25]. To present the result, we begin by introducing fourientations, which play a central role in Section 4.

The following observation generalizes Example 1(2):

Applying Lemma 5 to the positive fourientation \(\mathbf{F}_{+}\), we obtain Lemma 4 as a special case.

2.4 Ribbon graphs↩︎

A ribbon graph \(\mathbf{G}= (G,\Sigma)\) is a graph \(G = (V,E)\) together with a cellular embedding of \(G\) in a connected closed orientable⁹ surface \(\Sigma\), where a cellular embedding is an injective map the complement of whose image is homeomorphic to a disjoint union of open disks.

The vertices and edges of the ribbon graph \(\mathbf{G}\) are those of the graph \(G\).

The ribbon graph \(\mathbf{G}\) associated with \(G\hookrightarrow \Sigma\) naturally induces a dual graph \(G^* = (V^*,E^*)\). The vertices of \(G^*\) are points located in different components of the complement of \(G\), and each edge \(e^*\) of \(G^*\) traverses the corresponding edge \(e\) of the original graph \(G\). See Figure 1 for an example. The dual ribbon graph is \(\mathbf{G}^* := (G^*,\Sigma)\). We call the vertices and edges of \(G^*\) the covertices and coedges, respectively, of the ribbon graph \(\mathbf{G}\).

A transversal \(B\) of \(E\cup E^*\) is a basis of \(\mathbf{G}\) if \(\Sigma \setminus B\) is connected, and we denote by \(\mathcal{B}(\mathbf{G})\) the set of bases.

A subgraph \(H\) of \(G\) is spanning if its vertex set is \(V\), and is a quasi-tree if an \(\epsilon\)-neighborhood of \(H\) in \(\Sigma\) has exactly one boundary component for a sufficiently small \(\epsilon>0\). We often identify an edge set \(Q\subseteq E\) with the spanning subgraph \((V,Q)\), and let \(\mathcal{Q}(\mathbf{G}) \subseteq 2^E\) be the set of spanning quasi-trees of \(\mathbf{G}\).

By the following lemma, we can identify the spanning quasi-trees of \(\mathbf{G}\) with the bases of \(\mathbf{G}\).

We call \(M(\mathbf{G})\) a ribbon-graphic orthogonal matroid. By definition, we can characterize the circuits of \(M(\mathbf{G})\) as follows:

Every ribbon-graphic orthogonal matroid is regular, and the regularity is witnessed by the following representation.

First, by convention, fix the orientation on \(\Sigma\) which is counter-clockwise with respect to (outward) normal vectors. Every submanifold \(\Sigma'\) of \(\Sigma\) inherits an orientation from \(\Sigma\), and its boundary \(\partial \Sigma'\) is also endowed with an induced orientation. A precise definition of the induced orientation on a submanifold can be found in [26]. Informally, the induced orientation on \(\partial \Sigma'\) may be described as follows: if a person traverses \(\partial \Sigma'\) with their head pointing in the direction of the normal vector to \(\Sigma'\) and with \(\Sigma'\) on their left, then the induced orientation on \(\partial \Sigma'\) is the direction of travel.

Now choose a reference orientation \(\vec{G}\) of the set \(E\) of edges of \(G\), and assign the dual reference orientation to \(G^*\) by the “right-hand rule”, i.e., so that each coedge \(e^*\) crosses \(e\) from right to left (or, equivalently, \(e\) crosses \(e^*\) from left to right); see Figure 2.

Theorem 6 can be derived from the results of Booth, Borovik, Gelfand, and Stone [27], as described below. It also follows from earlier work of Bouchet [7], who proved a corresponding result in terms of circle graphs. We refer the reader to Appendix 7 for a comparison of his approach with that presented in this paper.

The intersection index \((p,q) \in \mathbb{Z}\) of two curves \(p\) and \(q\) on \(\Sigma\) is defined as the number of times \(p\) traverses \(q\) from left to right minus the number of times \(p\) traverses \(q\) from right to left. For example, \((e,e^*)=1\) for each edge \(e\). For a cycle \(c\) in the homology group \(H_1(\Sigma\setminus (V\cup V^*)) := H_1(\Sigma\setminus (V\cup V^*),\mathbb{Q})\), let \[\begin{align} \iota(c) := \sum_{e\in E\cup E^*} (c,e) e \in \mathbb{Q}^{E\cup E^*}. \end{align}\]

3 The Jacobian group and the circuit reversal classes↩︎

In this section, we first define the Jacobian group \(\mathrm{Jac}(\mathcal{C})\) and the circuit reversal classes \(\mathcal{R}(\mathcal{C})\) of a regular orthogonal representation \(\mathcal{C}\). Then we will prove that there is a canonical group action of \(\mathrm{Jac}(\mathcal{C})\) on the set \(\mathcal{R}(\mathcal{C})\).

3.1 Definition of the Jacobian group↩︎

Let \(\mathcal{C}\) be a regular orthogonal representation of an orthogonal matroid \(M=(E\cup E^*, \mathcal{B})\).

We define the function \[\pi\colon\mathbb{Q}^{E\cup E^*}\to\mathbb{Q}^E\] by \[\pi(\mathbf{v})(e)=\mathbf{v}(e)+\mathbf{v}(e^*).\]

Define \[\langle \pi(\mathcal{C})\rangle:=\text{the subgroup of }\mathbb{Z}^E\text{ generated by }\{\pi(\mathbf{C}):\mathbf{C}\in\mathcal{C}\}.\]

We remark that \(\pi(\mathbf{C})\in\{0, \pm 1\}^E\) for any signed circuit \(\mathbf{C}\), because the support of any signed circuit \(\mathbf{C}\) is a subtransversal.

We define an equivalence relation \(\sim_\mathrm{J}\) on \(\mathbb{Q}^E\) by the rule \[\mathbf{v}_1\sim_\mathrm{J}\mathbf{v}_2\Leftrightarrow \mathbf{v}_2-\mathbf{v}_1\in\langle \pi(\mathcal{C})\rangle.\] Note that \(\sim_\mathrm{J}\) induces an equivalence relation on \(\mathbb{Z}^E\). We denote an equivalence class for this relation by \([\mathbf{v}]_\mathrm{J}\).

3.2 The circuit reversal system↩︎

Our first main result of the paper is:

In order to prove Theorem 11, we must first show that the orientation \(\mathbf{O}_2\) in Definition 11 always exists. The proof uses Farkas’ Lemma for oriented orthogonal matroids (Lemma 4), which for the present purpose we reformulate as follows:

Define \[\mathbb{Z}_{\frac{1}{2}}^E:=\text{the set of functions from }E\text{ to }\{n+\frac{1}{2}:n\in\mathbb{Z}\}\] and note that \(\sim_\mathrm{J}\) induces an equivalence relation on \(\mathbb{Z}_{\frac{1}{2}}^E\). We again denote an equivalence class for this relation by \([\mathbf{v}]_\mathrm{J}\).

The next step is to show that the combinatorial equivalence relation \(\mathbf{O}_1\sim_\mathrm{R}\mathbf{O}_2\) coincides with the algebraic equivalence relation \(\mathbf{O}_2\sim_\mathrm{J}\mathbf{O}_1\). To prove this, we will need two more lemmas.

3.3 Invariance under reorientation↩︎

As before, let \(\mathcal{C}\) be a regular orthogonal representation of an orthogonal matroid \(M=(E\cup E^*, \mathcal{B})\). Fix a subset \(X\) of \(E\). Recall that the reorientation of \(\mathcal{C}\) on \(X\) is defined as \[\mathcal{C}^{X} := \{\mathbf{C}^{X} \in \mathbb{Z}^{E\cup E^*} : \mathbf{C}\in\mathcal{C}\},\] where \[\begin{align} \mathbf{C}^{X}(i) = \begin{cases} -\mathbf{C}(i) & \text{if } i\in X\cup X^*, \\ \mathbf{C}(i) & \text{otherwise}. \end{cases} \end{align}\]

Note also that \(\pi(\mathbf{C}^X)=(\pi(\mathbf{C}))^X\).

We will use the notation \([\mathbf{v}]_\mathrm{J}^X:=[\mathbf{v}^X]_\mathrm{J}\) and \([\mathbf{O}]_\mathrm{R}^X:=[\mathbf{O}^X]_\mathrm{R}\) to emphasize the fact that \([\mathbf{v}]_\mathrm{J}^X\in\mathrm{Jac}(\mathcal{C}^X)\) and \([\mathbf{O}]_\mathrm{R}^X\in\mathcal{R}(\mathcal{C}^X)\) respectively.

Putting together the various results proved in this section, we see that \(\mathrm{Jac}(\mathcal{C})\), \(\mathcal{R}(\mathcal{C})\), and the action of \(\mathrm{Jac}(\mathcal{C})\) on \(\mathcal{R}(\mathcal{C})\) all depend only on the regular orthogonal representation \(\mathcal{C}\) up to reorientation equivalence.

3.4 The lift of circuit reversals↩︎

The reader might have noticed that the signed circuits of a regular orthogonal representation \(\mathcal{C}\) are elements of \(\mathbb{Q}^{E \cup E^*},\) but the circuit reversal system is defined in terms of an equivalence relation on orientations, which are elements of \(\mathbb{Q}^E\). The map \(\pi : \mathbb{Q}^{E \cup E^*} \to \mathbb{Q}^E\) is used to compare the two sides. It is natural to wonder if one can “lift” orientations to \(\mathbb{Q}^{E \cup E^*}\) instead of “projecting” circuits to \(\mathbb{Q}^E\). In this section, we show that this is indeed the case.

Although the results of this section will not be used elsewhere in the paper, it seems worth recording this alternative description of the equivalence relation \(\sim_R\) generated by circuit reversals.

We now define a “lifting” map \(\psi : \mathcal{O}= \{\pm\frac{1}{2}\}^{E} \to \{0,\pm\frac{1}{2}\}^{E\cup E^*}\).

Note that the first two properties of \(\psi(\mathbf{O})\) imply that \[\psi(\mathbf{O})\in\{0,\pm\frac{1}{2}\}^{E\cup E^*}.\]

The next lemma follows immediately from Definition 14.

The following lemma gives an equivalent definition of \(\psi(\mathbf{O})\).

We can now give the following alternative description of circuit reversals.

As a direct consequence of Proposition 13, we obtain:

4 The BBY bijections and torsors of regular orthogonal representations↩︎

The BBY bijections between bases and circuit-cocircuit reversal classes of regular matroids are defined in [3]. In this section we define analogous bijections for regular orthogonal representations.

The proof of bijectivity in [3] involves zonotope tilings, and it is not clear how to define similar geometric objects for orthogonal matroids. We will therefore adopt the machinery introduced in [13], where the BBY bijections are generalized and the proof of bijectivity is purely combinatorial. We state and prove our general BBY bijections for regular orthogonal representations in Section 4.1. In Section 4.2, we introduce the notion of “acyclic signatures” and connect this to the results of the previous section. Finally, in Section 4.3 we define and study the corresponding BBY torsors. Throughout this section we fix a regular orthogonal representation \(\mathcal{C}\) of a regular orthogonal matroid \(M\) with ground set \(E\cup E^*\).

4.1 The BBY bijections↩︎

For the definition of fourientations and related notions, see Section 2.3.

Recall that orientations are elements in \(\mathcal{O}=\{\pm\frac{1}{2}\}^{E}\). For an orientation \(\mathbf{O}\in \mathcal{O}\) and an element \(e \in E\), we define\[\text{sign}(\mathbf{O}(e)) = \begin{cases} \{ 1\} & \text{if } \mathbf{O}(e)>0, \\ \{-1\} & \text{if } \mathbf{O}(e)<0.\\ \end{cases}\] We will use the following notation many times.

We now give the key definition in this section.

The following is the main theorem of this section.

4.2 Circuit signatures↩︎

In order to apply Theorem 15, one needs a criterion to certify that a given map \(\beta\colon\mathcal{B}\to\mathcal{O}\) is triangulating. As in the case of regular matroids [3], a convenient tool in this regard is the notion of acyclic signatures.

The following lemma is an immediate consequence of Definitions 15 and 19.

We will use the following property of acyclic circuit signatures to prove that \(\beta_\sigma\) is triangulating whenever \(\sigma\) is acyclic.

From Theorem 15 and Lemma 20, we obtain:

4.3 The BBY torsors↩︎

In Section 3, we have shown that the natural group action of \(\mathrm{Jac}(\mathcal{C})\) on \(\mathcal{R}(\mathcal{C})\) is simply transitive. And in Section 4.1, we have shown that the map \(\bar{\beta}\) from \(\mathcal{B}\) to \(\mathcal{R}(\mathcal{C})\) is a bijection whenever \(\beta\) is triangulating.

By composing the action and the bijection, we obtain, for every triangulating map \(\beta : \mathcal{B}\to \mathcal{O}\), a simply transitive action \(\Gamma_\beta\) of \(\mathrm{Jac}(\mathcal{C})\) on \(\mathcal{B}\) defined by\[\begin{align} \Gamma_\beta\colon\mathrm{Jac}(\mathcal{C}) \times \mathcal{B}&\to \mathcal{B}\\ ([\mathbf{v}]_\mathrm{J},B) &\mapsto \bar{\beta}^{-1}([\mathbf{v}]_\mathrm{J} \cdot \bar{\beta}(B)). \end{align}\]

This makes the set \(\mathcal{B}\) of bases of \(M\) into a torsor for the group \(\mathrm{Jac}(\mathcal{C})\).

Using Definition 11 and Definition 20, we have the following equivalent ways to describe a BBY torsor.

Our next goal is to show that the BBY bijections and BBY torsors are (in a suitable sense) invariant under reorientation.

It is straightforward to prove the following results.

The BBY torsor does not depend on reorientation in the following sense.

5 The canonical torsor for a ribbon graph↩︎

In the previous section, we have defined BBY torsors for regular orthogonal representations of regular orthogonal matroids. In this section, we apply the theory to ribbon graphs \(\mathbf{G}\). We obtain a canonical BBY torsor associated to \(\mathbf{G}\) by defining a natural class \(\{ \sigma_p \}\) of acyclic signatures and showing that they all give rise to the same torsor. We then define Bernardi-style bijections and Bernardi torsors for \(\mathbf{G}\), and we show that they coincide with the BBY bijections and BBY torsors for the acyclic signatures \(\sigma_p\). In particular, the Bernardi torsor is also canonical.

5.1 The Jacobian group and ribbon cycle reversal system of a ribbon graph↩︎

Let \(\mathbf{G}\) be a ribbon graph. From \(\mathbf{G}\), we obtain the orthogonal matroid \(M(\mathbf{G})\) whose bases are in bijection with the spanning quasi-trees of \(\mathbf{G}\) (cf. Lemma 6). Now choose a reference orientation \(\vec{G}\) of the set \(E\) of edges of \(\mathbf{G}\), and assign the dual reference orientation to \(E^*\) by the “right-hand rule”. From these data, we obtain the regular orthogonal representation \(\mathcal{C}(\mathbf{G},\vec{G})\) of \(M(\mathbf{G})\) (cf. Definition 8).

The objects defined in Definition 22 are independent of the choice of reference orientation \(\vec{G}\), in the following sense. To each edge \(e\) (resp. coedge \(e^*\)) of \(\mathbf{G}\) we associate two opposite directed edges (resp. coedges). The standard basis vectors \(\mathbf{e}\) of \(\mathbb{Z}^E=\oplus_{e\in E}\mathbb{Z}\mathbf{e}\) are interpreted as directed edges in \(\vec{G}\), and the opposite vectors \(-\mathbf{e}\) are interpreted as the opposite directed edges. The standard basis vectors \(\mathbf{e}^*\) of \(\mathbb{Z}^{E^*}=\oplus_{e^*\in E}\mathbb{Z}\mathbf{e}^*\) are interpreted similarly. A signed ribbon cycle \(\mathbf{C}\in\mathcal{C}(\mathbf{G},\vec{G})\) is interpreted as the set of directed edges and coedges whose sum is \(\mathbf{C}\). Then a different choice of \(\vec{G}\) yields a reorientation \(\mathcal{C}(\mathbf{G},\vec{G})^X\) of \(\mathcal{C}(\mathbf{G},\vec{G})\), where \(X\) is the set of edges where the two reference orientations differ (cf. Remark 8). Note that the signed ribbon cycles \(\mathbf{C}\in\mathcal{C}(\mathbf{G},\vec{G})\) and \(\mathbf{C}^X\in\mathcal{C}(\mathbf{G},\vec{G})^X\) are interpreted as the same set of directed edges and coedges. By Lemma 13, we may identify \([\mathbf{e}]_\mathrm{J}\in\mathrm{Jac}(\mathcal{C}(\mathbf{G},\vec{G}))\) with \([\mathbf{e}^X]_\mathrm{J}\in\mathrm{Jac}(\mathcal{C}(\mathbf{G},\vec{G})^X)\), and the two vectors \(\mathbf{e}\) and \(\mathbf{e}^X\) are interpreted as the same directed edge of \(\mathbf{G}\). In this sense, we may identify the Jacobian groups obtained from different reference orientations (and, in particular, these groups are isomorphic to one another).

Similarly, we may identify the ribbon cycle reversal systems \(\mathcal{R}(\mathcal{C}(\mathbf{G},\vec{G}))\) obtained from different reference orientations \(\vec{G}\) using Lemma 14. In particular, the circuit reversals \(\mathbf{O}+\pi(\mathbf{C})\) and \(\mathbf{O}^X+\pi(\mathbf{C}^X)\) have the same geometric interpretation: reverse the directions of the edges \(e\in\mathrm{supp}(\pi(\mathbf{C}))\) in \(\mathbf{O}\). By Lemma 15, the group action of \(\mathrm{Jac}(\mathcal{C}(\mathbf{G},\vec{G}))\) on \(\mathcal{R}(\mathcal{C}(\mathbf{G},\vec{G}))\) is independent of the choice of \(\vec{G}\).

The discussion above leads to a “geometric” description of the group action, without referring to the reference orientation \(\vec{G}\) (cf. [3]). More precisely, the action of \(\mathrm{Jac}(\mathbf{G})\) on \(\mathcal{R}(\mathbf{G})\) can be defined by linearly extending the following action of each generator \([\mathbf{e}]_\mathrm{J}\) of \(\mathrm{Jac}(\mathbf{G})\) on an orientation class \([\mathbf{O}]_\mathrm{R}\) of \(\mathcal{R}(\mathbf{G})\):

1. if the directed edge \(\mathbf{e}\) is not in \(\mathbf{O}\), then reverse the orientation of \(e\) in \(\mathbf{O}\) to obtain \([\mathbf{e}]_\mathrm{J}\cdot[\mathbf{O}]_\mathrm{R}\);

2. otherwise (using Lemma 9), find a signed ribbon cycle compatible with \(\mathbf{O}\), reverse the cycle in \(\mathbf{O}\), and then apply (i).

It is straightforward to check that this definition is equivalent to Definition 11 for ribbon graphs.

5.2 The canonical BBY torsor for a ribbon graph↩︎

Recall that the ribbon graph \(\mathbf{G}\) is embedded into a connected closed orientable surface \(\Sigma\).

For simplicity, we denote \[\beta_p:=\beta_{\sigma_p}.\] By Proposition 19 and Corollary 6, the map \(\bar{\beta}_p\) is a BBY bijection between the set \(\mathcal{Q}(\mathbf{G})\) of spanning quasi-trees of \(\mathbf{G}\) and the set \(\mathcal{R}(\mathbf{G})\) of ribbon cycle reversal system of \(\mathbf{G}\). Hence we obtain a simply transitive group action \(\Gamma_p\) of the Jacobian group \(\mathrm{Jac}(\mathbf{G})\) on \(\mathcal{Q}(\mathbf{G})\), given by the formula (cf. Section 4.3) \[\begin{align} \Gamma_p\colon\mathrm{Jac}(\mathbf{G}) \times \mathcal{Q}(\mathbf{G}) &\to \mathcal{Q}(\mathbf{G}) \\ ([\mathbf{v}]_\mathrm{J},B) &\mapsto \bar{\beta}_p^{-1}([\mathbf{v}]_\mathrm{J} \cdot \bar{\beta}_p(B)), \end{align}\] where we identify a spanning quasi-tree \(Q\in\mathcal{Q}\) with the basis \(B=Q\cup (E\setminus Q)^*\).

Although the bijection \(\beta_p\) depends on the choice of \(p\) in general, we will show that the group action \(\Gamma_p\) does not.

By Corollary 7 and Theorem 20, the action \(\Gamma := \Gamma_p\) depends only on the ribbon graph \(\mathbf{G}\) and our convention for how to form a dual orientation. In this sense, we say that the action \(\Gamma\) is canonical.

To prove the theorem, we start with a lemma. This lemma follows from Bouchet’s results on circle graphs [7] and multimatroids [16], but we give a direct proof here for completeness.

For a given edge \(e\in E\), let \[\delta_e:E\to\mathbb{Z}\] be the function that sends \(e\) to \(1\) and the other edges to \(0\).

Using the same method, we may prove the following result.

5.3 The Bernardi bijections for ribbon graphs↩︎

We extend the Bernardi bijections on graphs [2], [28] to ribbon graphs, and we show that they can be identified with a special subclass of the BBY bijections introduced in Section 5.2.

Let \(\mathbf{G}= (G,\Sigma)\) be a ribbon graph. In this subsection, we do not assume a fixed reference orientation of \(G\) a priori. This approach ensures that our result is as general as possible while maintaining compatibility with the original result by Bernardi [28].

We denote by \(v_e\) the unique point in the intersection of an edge \(e\) and the corresponding coedge \(e^*\), so that \(v_e\) divides \(e\) (resp. \(e^*\)) into two half-edges (resp. half-coedges). Given a half-edge \(h\) associated with an edge \(e\), let \(h^*\) be the half-coedge that immediately follows \(h\) in the counter-clockwise ordering among the four half-(co)edges incident with \(v_e\); see Figure 6. We also define \((h^*)^* := h\), so that \(*\) is an involution on the set of half-edges and half-coedges. In particular, the operation \(*\) rotates half-coedges clockwise.

Let \(N_\epsilon\) be a small \(\epsilon\)-neighborhood of a spanning quasi-tree \(Q \subseteq E\). For \(e\in E\cup E^*\), the boundary \(\partial N_\epsilon\) intersects \(e\) if and only if \(e \in Q^* \cup (E\setminus Q)\). Whenever \(\partial N_\epsilon\) intersects \(e\), it traverses each of the corresponding half-(co)edges exactly once. Consequently, we obtain a cyclic ordering, denoted \(Q^\circ\), on the \(2|E|\) half-(co)edges that intersect \(\partial N_\epsilon\) by enumerating them according to the induced orientation on \(\partial N_\epsilon\). See Example 12.

We consider a pair \((h_1,h_2)\) of half-edges associated with an edge \(e\) as a directed edge \(\vec{e}\) with tail \(h_1\) and head \(h_2\).

Now fix a reference orientation \(\vec{G}\), and for a half-edge \(h\), let \(\beta_h'(-,\vec{G}) : \mathcal{B}(\mathbf{G}) \to \mathcal{O}= \{\pm\frac{1}{2}\}^E\) be the map defined by \[\begin{align} \beta_h'(B,\vec{G})(e) = \begin{cases} +\frac{1}{2} & \text{if the directions of e are the same in \vec{G} and \gamma_h(E\cap B)}, \\ -\frac{1}{2} & \text{otherwise}. \end{cases} \end{align}\]

Let \(p\) be the point of \(\Sigma \setminus (G\cup G^*)\) which is located at distance \(\epsilon/2\) from the midpoint of \(h\) in the direction of \(h^*\); see Figure 7. We denote by \(\beta_p(-,\vec{G})\) the corresponding BBY bijection.

6 Complements↩︎

6.1 Orientation conventions↩︎

Our theory for ribbon graphs \(\mathbf{G}=(G,\Sigma)\) has been built on the convention that the orientation of the surface \(\Sigma\) is counter-clockwise and the dual reference orientation is assigned using the right-hand rule. If we instead use the clockwise orientation of \(\Sigma\) and the left-hand rule for the dual orientation, this affects the definition of \(\sigma_p\) (cf. Definition 23) and the cyclic ordering used to define the Bernardi map \(\beta_h'\) (cf. Definition 24). With these alternative conventions, Theorem 23 can be stated as \[\mathfrak{b}_h'(-,\vec{G})=\mathfrak{b}_p(-,\vec{G}),\] where the Bernardi map \(\mathfrak{b}_h'\) and the BBY map \(\mathfrak{b}_p\) are defined using the clockwise orientation of \(\Sigma\) and the left-hand rule, and \(\vec{G}\) is a reference orientation on \(E\) independent of the choice of convention.

6.2 Duality↩︎

We explain how the canonical BBY torsor defined in Section 4 (or equivalently the Bernardi torsor defined in Section 5) is compatible with duality:

Given an orthogonal matroid \(M=(E\cup E^*, \mathcal{B})\), its dual orthogonal matroid is defined to be\[M^*=(E\cup E^*, \mathcal{B}^*),\] where\[\mathcal{B}^*=\{B^*:B\in\mathcal{B}\}.\] This generalizes the notion of dual matroids because for any matroid \(N\), we have \(\mathrm{lift}(N)^*=\mathrm{lift}(N^*)\). Clearly, \((M^*)^*=M\).

It is straightforward to check that the set \(\mathcal{C}^*\) satisfies Definition 3. Because for any \(\mathbf{C}\in\mathcal{C}\) we have \(\pi(\mathbf{C})=\pi(\mathbf{C}^*)\), the identity map induces

• the group isomorphism \(\mathrm{Jac}(\mathcal{C})\cong\mathrm{Jac}(\mathcal{C}^*)\),

• the bijection \(\mathcal{R}(\mathcal{C})\to\mathcal{R}(\mathcal{C}^*)\), and

• the “isomorphic actions” \(\mathrm{Jac}(\mathcal{C})\circlearrowright\mathcal{R}(\mathcal{C})\) and \(\mathrm{Jac}(\mathcal{C}^*)\circlearrowright\mathcal{R}(\mathcal{C}^*)\).

We will identify the two objects in each pair.

Now we consider the BBY bijections. Given an acyclic circuit signature \(\sigma\) of \(M\), the set \[\sigma^*:=\{\mathbf{C}^*:\mathbf{C}\in\sigma\}\] is clearly an acyclic circuit signature of \(M^*\). Note that for any basis \(B\) of \(M\) and any \(e\notin B\), we have \(\mathrm{FC}(B^*,e^*)=\mathrm{FC}(B,e)^*\). Therefore, we have \[\label{eq:32dual32BBY} \beta_\sigma(B)=\beta_{\sigma^*}(B^*).\tag{3}\]

Next we study the dual of a ribbon graph \(\mathbf{G}\) with edge set \(E\). As defined in Section 2.4, the dual ribbon graph \(\mathbf{G}^*\) has edge set \(E^*\), but in this subsection we interchange \(E\) and \(E^*\) so that we can consider spanning quasi-trees of \(\mathbf{G}^*\) as subsets of \(E\). Clearly, we have:

• \(M(\mathbf{G})^*=M(\mathbf{G}^*)\),

• a subset \(Q\) of \(E\) is a spanning quasi-tree of \(\mathbf{G}\) if and only if \(E\setminus Q\) is a spanning quasi-tree of \(\mathbf{G}^*\), and

• a subset \(C\) of \(E\cup E^*\) is a circuit of \(M(\mathbf{G})\) if and only if \(C^*\) is a circuit of \(M(\mathbf{G}^*)\).

We remark that a circuit \(C\) of \(M(\mathbf{G})\) is geometrically the same as the circuit of \(C^*\) of \(M(\mathbf{G}^*)\); see Figure 8 for example.

Recall that the regular orthogonal representation \(\mathcal{C}(\mathbf{G}, \vec{G})\) is obtained from the ribbon graph \(\mathbf{G}\), a reference orientation \(\vec{G}\) of \(E\), and the dual reference orientation \(\vec{G}^*\) of \(E^*\) induced by the right-hand rule. To make our torsors compatible with duality, when we define the regular orthogonal representation \(\mathcal{C}(\mathbf{G}^*, \vec{G}^*)\) for \(M(\mathbf{G}^*)\), we use the left-hand rule (and the clockwise orientation of \(\Sigma\)), where \(\vec{G}^*\) is the reference orientation of \(E^*\) for \(\mathbf{G}\) and hence a reference orientation of \(E\) for \(\mathbf{G}^*\); see Figure 8. An immediate consequence is that any signed ribbon cycle \(\mathbf{C}\) of \(\mathbf{G}\) is the same geometric object and the same algebraic object as the signed ribbon cycle \(\mathbf{C}^*\) of \(\mathbf{G}^*\). In particular, we have the identity\[\mathcal{C}(\mathbf{G}^*, \vec{G}^*)=\mathcal{C}(\mathbf{G}, \vec{G})^*\]and the isomorphism\[\mathrm{Jac}(\mathbf{G})\cong\mathrm{Jac}(\mathbf{G}^*)\] induced by the identity map from \(\mathbb{Z}^E\) to \(\mathbb{Z}^E\).

Fix a point \(p\in \Sigma\setminus(E\cup E^*)\). By Equation (3 ), we get\[\beta_{\sigma_p}(B)=\beta_{\sigma_p^*}(B^*),\] where \(\sigma_p\) is defined in Definition 23 for the ribbon graph \(\mathbf{G}\). Because we use the left-hand rule for \(\mathbf{G}^*\) along with the clockwise surface orientation, we have \[\beta_{\sigma_p^*}(B^*)=-\mathfrak{b}_{p}(B^*,\vec{G}^*).\] Applying Theorem 23 to \(\mathbf{G}\) and its left-hand analogue (cf. Section 6.1) to \(\mathbf{G}^*\), we get \[\beta_h'(B,\vec{G})=\beta_p(B,\vec{G})(=\beta_{\sigma_p}(B))\] and \[\mathfrak{b}_h'(B^*,\vec{G}^*)=\mathfrak{b}_p(B^*,\vec{G}^*).\]The four identities above imply \[\label{eq:32dual32relation} \beta_h'(B,\vec{G})=\beta_p(B,\vec{G})=-\mathfrak{b}_{p}(B^*,\vec{G}^*)=-\mathfrak{b}_h'(B^*,\vec{G}^*).\tag{4}\]

Recall that for the ribbon graph \(\mathbf{G}\) (and the right-hand rule), we have the canonical group action \[\Gamma_p\colon\mathrm{Jac}(\mathbf{G}) \times \mathcal{Q}(\mathbf{G}) \to \mathcal{Q}(\mathbf{G})\] determined by the relation \[\Gamma_p([\mathbf{v}]_\mathrm{J},B_1)=B_2 \Leftrightarrow [\mathbf{v}]_\mathrm{J}\cdot\bar{\beta}_p(B_1,\vec{G})=\bar{\beta}_p(B_2,\vec{G}).\] For the dual ribbon graph \(\mathbf{G}^*\) (and the left-hand rule), the canonical group action\[\Gamma_p^*\colon\mathrm{Jac}(\mathbf{G}^*) \times \mathcal{Q}(\mathbf{G}^*) \to \mathcal{Q}(\mathbf{G}^*)\]is determined by the relation \[\Gamma_p^*([\mathbf{v}]_\mathrm{J},B_1^*)=B_2^* \Leftrightarrow [\mathbf{v}]_\mathrm{J}\cdot\bar{\mathfrak{b}}_p(B_1^*,\vec{G}^*)=\bar{\mathfrak{b}}_p(B_2^*,\vec{G}^*).\]

By Equation (4 ), the two group actions satisfy the following property, which means that the canonical torsor \(\Gamma_p\) is compatible with duality.

This generalizes [2] which shows that the Bernardi torsor for a plane graph is compatible with duality.

6.3 Connections to regular matroids↩︎

We explain how our definitions and results generalize the corresponding ones for regular matroids.

Recall that given a matroid \(N = (E,\mathcal{B})\), the pair \(\mathrm{lift}(N):=(E\cup E^*, \mathcal{B}')\) is an orthogonal matroid, where \(\mathcal{B}' = \{B\cup (E\setminus B)^* : B\in \mathcal{B}\}\). The set of circuits of \(\mathrm{lift}(N)\) is the union of \(\{C\subseteq E: \text{C is a circuit of N}\}\) and \(\{D^*\subseteq E^*: \text{D is a cocircuit of N}\}\).

Let \(N\) be a regular matroid. Choosing a regular representation of \(N\) (i.e., a totally unimodular integer matrix representing \(N\)), we obtain a set \(\mathcal{C}\) of signed circuits of \(N\) and a set \(\mathcal{D}\) of signed cocircuits of \(N\), where \(\mathcal{C},\mathcal{D}\subseteq\{0,\pm 1\}^E\) and every signed circuit is orthogonal to every signed cocircuit with respect to the standard inner product on \(\mathbb{Z}^E\). See [3] for details. The sets \(\mathcal{C}\) and \(\mathcal{D}\) are unique up to reorientations; this result is essentially due to [29].

For every element \(\mathbf{v}\in\mathcal{C}\) (resp. \(\mathbf{v}\in\mathcal{D}\)), let \(\phi(\mathbf{v})\in\{0,\pm 1\}^{E\cup E^*}\) be the unique vector such that \(\mathrm{supp}(\phi(\mathbf{v}))\subseteq E\) (resp. \(\mathrm{supp}(\phi(\mathbf{v}))\subseteq E^*\)) and \(\pi(\phi(\mathbf{v}))=\mathbf{v}\). Denote\[\mathcal{C}\mathcal{D}:=\{\phi(\mathbf{v}):\mathbf{v}\in\mathcal{C}\cup\mathcal{D}\}\subseteq\{0,\pm 1\}^{E\cup E^*}.\] The set \(\mathcal{C}\mathcal{D}\) is a regular orthogonal representation of \(\mathrm{lift}(N)\), as one can easily check.

Recall that the Jacobian group of \(N\) is defined to be \[\mathrm{Jac}(N):=\mathbb{Z}^E/\langle\mathcal{C}\cup\mathcal{D}\rangle.\]

We recall the circuit-cocircuit reversal system for regular matroids, defined in [10], using our language. The set of orientations is \(\{\pm \frac{1}{2}\}^E\). A circuit reversal (resp. a cocircuit reversal) of the regular matroid \(N\) turns an orientation \(\mathbf{O}\) into the orientation \(\mathbf{O}+\mathbf{C}\) for some \(\mathbf{C}\in\mathcal{C}\) (resp. for some \(\mathbf{C}\in\mathcal{D}\)). Two orientations \(\mathbf{O}_1\) and \(\mathbf{O}_2\) of \(N\) are said to be equivalent if one orientation can be obtained from the other by a sequence of circuit reversals and cocircuit reversals. This defines an equivalence relation on the set of orientations. We call an equivalence class for this relation a circuit-cocircuit reversal class. The set of circuit-cocircuit reversal classes of \(N\) is denoted by \(\mathcal{R}(N)\). Because a circuit reversal or a cocircuit reversal \(\mathbf{O}+\mathbf{C}\) of the regular matroid \(N\) is the same as a circuit reversal \(\mathbf{O}+\pi(\phi(\mathbf{C}))\) of the regular orthogonal representation \(\mathcal{C}\mathcal{D}\), we have the following result:

Similarly, one can check that the following definitions for the regular orthogonal representation \(\mathcal{C}\mathcal{D}\) are essentially the same as their counterparts for regular matroids. We omit the proofs for brevity.

Acknowledgements↩︎

Thanks to Nathan Bowler, Matt Larson, Langte Ma, Iain Moffatt, and Wei Wang for helpful discussions, and to Chi Ho Yuen for feedback on the first draft of this manuscript. This work began at the 2024 Workshop on (Mostly) Matroids at the Institute for Basic Science, Daejeon, South Korea, in August 2024, and we would like to thank IBS as well as the organizers of that worshop (S. Backman, R. Campbell, D. Mayhew, and S. Oum) for providing a stimulating environment. The first author was supported by NSF grant DMS-2154224 and a Simons Fellowship in Mathematics. The third author was supported by the Institute for Basic Science (IBS-R029-C1).

7 Bouchet’s representations of ribbon graphs↩︎

In this appendix, we review Bouchet’s construction [7] of principally unimodular matrices associated with ribbon graphs, with the ultimate goal of proving that the Jacobian group \(\mathrm{Jac}(\mathbf{G})\), as defined in this paper, coincides with the critical group of \(\mathbf{G}\), as defined in [5]. Let \(\mathbf{G}= (G,\Sigma)\) be a ribbon graph, and fix a reference orientation of \(G\). We denote by \(v_e\) the intersection of an edge \(e\) and the corresponding coedge \(e^*\). The point \(v_e\) divides \(e\) into the two half-edges \(e_{+}\) and \(e_{-}\), where we adopt the convention that \(e_{+}\) corresponds to the head of \(e\) and \(e_{-}\) corresponds to the tail. Similarly, we define the two half-coedges \(e^*_{+}\) and \(e^*_{-}\).

We define a matrix \(A(\mathbf{G},Q)\) as follows:

1. Select a spanning quasi-tree \((V,Q)\) of \(\mathbf{G}\). We denote by \(q\) the boundary of a small \(\epsilon\)-neighborhood of \((V,Q)\) in \(\Sigma\). Then, as in §5.3, \(q\) intersects \(2|E|\) half-edges and half-coedges induced by \(Q^* \cup (E\setminus Q)\).

2. As in §5.3, travel along \(q\) according to the induced orientation and list the \(2|E|\) half-edges and half-coedges you cross, in order; this yields a cyclic ordering which we denote by \(q^\circ\).

3. Form the skew-symmetric matrix \(A(\mathbf{G},Q)\), whose rows and columns are indexed by the elements in \(Q^* \cup (E\setminus Q)\), using the rule: \[\begin{align} A(\mathbf{G},Q)(e,f) = \begin{cases} +1 & \text{if we have \cdots f_{+} \cdots e_{+} \cdots f_{-} \cdots e_{-} \cdotsin q^\circ}, \\ -1 & \text{if we have \cdots e_{+} \cdots f_{+} \cdots e_{-} \cdots f_{-} \cdotsin q^\circ}, \\ 0 & \text{otherwise}. \\ \end{cases} \end{align}\]

The skew-symmetric matrix \(A(\mathbf{G},Q)\) encodes the signed ribbon cycles \(\mathcal{C}(\mathbf{G})\) of \(\mathbf{G}\), in the following sense. Let \(\Lambda(\mathbf{G},Q)\) be the \(n\times 2n\) matrix \(\begin{pmatrix} I & A(\mathbf{G},Q) \end{pmatrix}\) such that the columns of the identity matrix \(I\) are indexed by elements in \(Q\cup (E\setminus Q)^*\) and \(I(e,e^*) = 1\) for each \(e\in Q^*\cup (E\setminus Q)\).

Proposition 29, combined with Lemma 3, shows that \[\begin{align} \label{eq:criticalgroup} \mathrm{Jac}(\mathbf{G}) = \mathbb{Z}^E / \langle I + A(\mathbf{G},Q) \rangle \end{align}\tag{5}\] for any spanning quasi-tree \(Q\). We note that the right-hand side of 5 is the definition of “critical group” in [5].

References↩︎

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1. Also referred to in the literature as the critical group, sandpile group, or Picard group of \(G\).↩︎

2. Intuitively, when a group \(G\) acts simply and transitively on a set \(X\), we can think of \(X\) as being akin to a “group without a distinguished identity element”. This is made precise by the theory of torsors. More formally, given a group \(G\), a \(G\)-torsor is a set \(X\) on which \(G\) acts simply and transitively; for example, \(G\) is itself a \(G\)-torsor via left multiplication. Once we fix a reference point in a \(G\)-torsor \(X\), there is a natural isomorphism of \(G\)-torsors between \(X\) and \(G\) itself.↩︎

3. Technically speaking, we need \(G\) to be cellularly embedded, see §2.4 for the definition.↩︎

4. Ribbon graphs are also known in the literature as combinatorial maps or rotation systems.↩︎

5. Bouchet works with the equivalent concept of even delta-matroids, rather than orthogonal matroids, and many authors (e.g. Vince and White in [8]) call them Lagrangian orthogonal matroids; we omit the adjective “Lagrangian” since we will not consider the non-Lagrangian case. Recall also that we are assuming the ambient surface \(\Sigma\) is orientable; without this assumption Bouchet’s construction still yields a delta-matroid, but it will no longer be even.↩︎

6. The dependence of \(\mathrm{Jac}(M,[\mathcal{C}])\) on \([\mathcal{C}]\) clarifies the observation in [5] that \(\mathrm{Jac}(\mathbf{G})\) does not depend only on the even delta-matroid associated to \(\mathbf{G}\). When \(M\) arises from a regular matroid, there is a unique equivalence class of regular representations, but for more general regular orthogonal matroids this is no longer true.↩︎

7. The notion of circuit-cocircuit reversal system for graphs (resp. regular matroids) is due to Gioan [9] (resp. [10]). The circuit-cocircuit reversal system for ribbon graphs (resp. regular orthogonal matroids) is new to this paper.↩︎

8. Subtransversals are also known in the literature as admissible sets.↩︎

9. Ribbon graphs are generally defined on arbitrary closed surfaces, regardless of orientability. However, we assume in this paper that the ambient surface for every ribbon graph is orientable.↩︎

10. Our notion of the Jacobian group is isomorphic to the critical group of \(\mathbf{G}\) as defined by Merino, Moffatt, and Noble [5]; see Appendix 7 for a proof.↩︎

11. Technically speaking, we use a different convention. The action \([\mathbf{v}]_\mathrm{J}\cdot [\mathbf{O}]_\mathrm{R}\) in our paper is the same as the action \([-\mathbf{v}]_\mathrm{J}\cdot [\mathbf{O}]_\mathrm{R}\) in [3].↩︎