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

Overview & Synopsis

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 allow 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 slit along it, slide one notch, and sew 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 case of unicellular toroidal maps, the simple structure of noncontractible loops makes it possible to fully relate general maps to bipartite maps.