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The sandpile group of a plane graph has a canonical torsor structure on the spanning trees. This torsor can either be defined via rotor-routing or via the Bernardi process, or via trinities. It is most often called the rotor-routing torsor.
Klivans conjectured that for a plane graph, this torsor is in some sense the unique torsor of the sandpile group on the spanning trees. This conjecture was made precise by Ganguly and McDonough, who proposed the notion of conistency for a sandpile torsor. Consistency means that the torsor behaves well with respect to deletion and contraction of the edges of the graph. Then they showed that the rotor-routing torsor of a plane graph is consistent, moreover, it is the unique consistent sandpile torsor for plane graphs.
In their proofs, they used the rotor-routing definition for the torsor. In this note, we give an alternative, somewhat simpler proof for the consistency of the action using the trinity definition of the rotor-routing action.
We also give an example highlighting that considering adjacent vertices in the definition of consistency is important.
Bibtex entry:
| AUTHOR | = | {T{\'o}thm{\'e}r{\'e}sz, Lilla}, |
| TITLE | = | {Consistency of the planar rotor-routing action via the trinity definition}, |
| NOTE | = | {{\tt egres.elte.hu}}, |
| INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
| YEAR | = | {2023}, |
| NUMBER | = | {QP-2023-01} |