Determining Genus From Sandpile Torsor Algorithms
Alex McDonough
TL;DR
This paper demonstrates that rotor routing and Bernardi sandpile torsors do not uniquely determine a graph's genus, but the genus can be identified if the ribbon structure is known, resolving a question by M. Chan.
Contribution
It provides examples of graphs with identical sandpile torsors but different genus, and shows how to determine genus from sandpile torsors with known ribbon structure.
Findings
Constructed ribbon graphs with same torsors but different genus
Proved genus can be determined from torsors if ribbon structure is known
Resolved a question posed by M. Chan
Abstract
We provide a pair of ribbon graphs that have the same rotor routing and Bernardi sandpile torsors, but different topological genus. This resolves a question posed by M. Chan [Cha]. We also show that if we are given a graph, but not its ribbon structure, along with the rotor routing sandpile torsors, we are able to determine the ribbon graph's genus.
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