Enumeration of $r$-regular Maps on the Torus. Part II: Enumeration of Unsensed Maps

TL;DR

This paper develops a method to count $r$-regular maps on the torus up to all homeomorphisms, including orientation-reversing symmetries, by analyzing quotient maps on related surfaces like the Klein bottle and M"obius band.

Contribution

It introduces a detailed enumeration technique for unsensed $r$-regular maps on the torus considering all homeomorphisms, including orientation-reversing ones.

Findings

Derived recurrence relations for counting maps

Enumerated $r$-regular maps for various $r$

Connected quotient maps on surfaces with boundary

Abstract

The second part of the paper is devoted to enumeration of $r$-regular toroidal maps up to all homeomorphisms of the torus (unsensed maps). We describe in detail the periodic orientation reversing homeomorphisms of the torus which turn out to be representable as glide reflections. We show that considering quotients of the torus with respect to these homeomorphisms leads to maps on the Klein bottle, annulus and the M\"obius band. Using $3$- and $4$-regular maps as an example we describe the technique of enumerating quotient maps on surfaces with a boundary. Obtained recurrence relations are used to enumerate unsensed $r$-regular maps on the torus for various $r$.