The Jacobian of a regular orthogonal matroid and torsor structures on spanning quasi-trees of ribbon graphs

Abstract

Previous work of Chan--Church--Grochow and Baker--Wang shows that the set of spanning trees in a plane graph $G$ is naturally a torsor for the Jacobian group of $G$. Informally, this means that the set of spanning trees of $G$ naturally forms a group, except that there is no distinguished identity element. We generalize this fact to graphs embedded on orientable surfaces of arbitrary genus, which can be identified with ribbon graphs. In this generalization, the set of spanning trees of $G$ is replaced by the set of spanning quasi-trees of the ribbon graph, and the Jacobian group of $G$ is replaced by the Jacobian group of the associated regular orthogonal matroid $M$ (along with an associated regular representation of $M$). Our proof shows, more generally, that the family of "BBY torsors" constructed by Backman--Baker--Yuen and later generalized by Ding admit natural generalizations to (regular representations of) regular orthogonal matroids. In addition to shedding light on the role of planarity in the earlier work mentioned above, our results represent one of the first substantial applications of orthogonal matroids (also called "even delta-matroids" or "Lagrangian orthogonal matroids") to a natural combinatorial problem about graphs.

Figures (9)

• Figure 1: A graph $G = (\{v,w\},\{1,2,3,4\})$ embedded on a torus. The graph $G$ is colored black and its dual $G^*$ is colored red.
• Figure 2: A ribbon graph with an orientation and dual orientation (left), and a connected component $\Sigma'$ of $\Sigma$ minus edges $3$ and $4$ (right). The induced orientation on $\partial \Sigma'$ is colored green.
• Figure 3: An example for the proof of Lemma \ref{['lem:vC and fudamental circuit']}, applied to the ribbon graph in Figure \ref{['fig:orientation']}, $B=1234^*$, and $x=4$. The two boundaries $c$ and $c'$ of a small $\epsilon$-neighborhood of $(V, (E\cap B)\triangle\{x\})$ are colored by green and orange, respectively. Note that $\mathrm{FC}(B,x) = 34$.
• Figure 4: The point $p$ is near the intersection of the edge $1$ and the coedge $1^*$, under the edge $1$ and to the left of the coedge $1^*$. The arrows indicate the reference orientation.
• Figure 5: The points $p$, $q$, $r$, and $s$ defined in Lemma \ref{['lem: independent p']} and Lemma \ref{['lem: independent p full version']}.

Theorems & Definitions (157)

• Theorem A
• Theorem
• Theorem B
• Definition 2.1
• Theorem 2.2: BMP2003
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Lemma 2.6