Link between bipartite and general unicellular toroidal maps via slit--slide--sew bijections

TL;DR

This paper introduces a bijection linking general and bipartite unicellular toroidal maps using slit-slide-sew operations, providing a comprehensive understanding of their relationship, especially for genus 1 maps with a single face.

Contribution

It develops a novel bijection method for relating general and bipartite maps on toroidal surfaces, expanding the combinatorial understanding of these structures.

Findings

Established an involution on genus maps with even degree faces.

Provided a full interpretation of the relation between general and bipartite maps for genus 1.

Utilized rotations along specific loops to modify map structures.

Abstract

We relate general maps to bipartite maps through a bijection of type slit-slide-sew. We provide an involution on arbitrary genus maps with even degree faces. This enables a full interpretation of the relation between general and bipartite maps, in the case of genus $1$ maps with a unique face. The main tool is the use of rotations along well-chosen specific loops. Once a noncontractible simple loop is given, one slits along it, slides one notch, and sews back. This mildly modifies the structure of the map along the loop, changing the parity of the length of other loops crossing it. In the unicellular toroidal setting, the structure of noncontractible loops is simple enough to enable a full correspondence between general and bipartite maps.