Related papers: Determining Genus From Sandpile Torsor Algorithms
Let $G$ be a ribbon graph. Matthew Baker and Yao Wang proved that the rotor-routing torsor and the Bernardi torsor for $G$, which are two torsor structures on the set of spanning trees for the Picard group of $G$, coincide when $G$ is…
We make precise and prove a conjecture of Klivans about actions of the sandpile group on spanning trees. More specifically, the conjecture states that there exists a unique ``suitably nice'' sandpile torsor structure on plane graphs which…
Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing…
Baker and Wang define the so-called Bernardi action of the sandpile group of a ribbon graph on the set of its spanning trees. This potentially depends on a fixed vertex of the graph but it is independent of the base vertex if and only if…
The sandpile group Pic^0(G) of a finite graph G is a discrete analogue of the Jacobian of a Riemann surface which was rediscovered several times in the contexts of arithmetic geometry, self-organized criticality, random walks, and…
We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.
We prove that the genus of the Turaev surface of a link diagram is determined by a graph whose vertices correspond to the boundary components of the maximal alternating regions of the link diagram. Furthermore, we use these graphs to…
We study two actions of the (degree 0) Picard group on the set of the spanning trees of a finite ribbon graph. It is known that these two actions, denoted $\beta_q$ and $\rho_q$ respectively, are independent of the base vertex $q$ if and…
Sandpile groups are a subtle graph isomorphism invariant, in the form of a finite abelian group, whose cardinality is the number of spanning trees in the graph. We study their group structure for graphs obtained by attaching a cone vertex…
Let G be a ribbon graph, i.e., a connected finite graph G together with a cyclic ordering of the edges around each vertex. By adapting a construction due to O. Bernardi, we associate to any pair (v,e) consisting of a vertex v and an edge e…
Higher genus modular graph tensors map Feynman graphs to functions on the Torelli space of genus-$h$ compact Riemann surfaces which transform as tensors under the modular group $Sp(2h , \mathbb Z)$, thereby generalizing a construction of…
The work provides a brief intuitive overview theory of graph on surfaces. We considers graphs with an additional structure, wich we call discs with ribbons, also known as one-vertex ribbon graphs. And solves the problem (Skopenkov's) about…
Partial duality is a duality of ribbon graphs relative to a subset of their edges generalizing the classical Euler-Poincare duality. This operation often changes the genus. Recently J.L.Gross, T.Mansour, and T.W.Tucker formulated a…
In this paper, we analyze embeddings of grid graphs on orientable surfaces. We determine the genus of a large class of k-dimensional grid graphs and effective two-sided bounds for the genus of any 3-dimensional grid graph, both in terms of…
In this purely experimental work we try to represent the set of plane maps with 3 vertices and 3 faces as a bipartite ribbon graph. In particular, this construction allows one to estimate the genus of the initial set.
The genus of a graph is a topological invariant that measures the minimum genus of a surface on which the graph can be embedded without any edges crossing. Graph genus plays a fundamental role in topological graph theory, used to classify…
The aim of the current work is to investigate structural properties of the sandpile group of a special class of self-similar graphs. More precisely, we consider Abelian sandpiles on Sierpinski gasket graphs and for the choice of normal…
We study a current transport through a system of a few grains connected with tunneling links. The exact solution is given for an arbitrarily connected double-grain system with a shared gate in the framework of the orthodox model. The…
In the first part of our paper, we show that there exist non-isomorphic derived equivalent genus $1$ curves, and correspondingly there exist non-isomorphic moduli spaces of stable vector bundles on genus $1$ curves in general. Neither…
We examine connections between the gonality, treewidth, and orientable genus of a graph. Especially, we find that hyperelliptic graphs in the sense of Baker and Norine are planar. We give a notion of a bielliptic graph and show that each of…