Maps, immersions and permutations

TL;DR

This paper develops a permutation-based method to count and classify topologically distinct 4-valent planar maps and circle immersions in spheres and higher genus surfaces, extending previous results and considering symmetries and bicolourability.

Contribution

It introduces a novel permutation encoding approach for counting and listing inequivalent maps and immersions, including bicoloured and higher genus cases, with detailed symmetry considerations.

Findings

Extended enumeration of spherical curves and maps with symmetry classifications.

Method applies to higher genus surfaces and distinguishes bicolourable from general maps.

Provides explicit counting formulas using permutation group actions.

Abstract

We consider the problem of counting and of listing topologically inequivalent "planar" {4-valent} maps with a single component and a given number n of vertices. This enables us to count and to tabulate immersions of a circle in a sphere (spherical curves), extending results by Arnold and followers. Different options where the circle and/or the sphere are/is oriented are considered in turn, following Arnold's classification of the different types of symmetries. We also consider the case of bicolourable and bicoloured maps or immersions, where faces are bicoloured. Our method extends to immersions of a circle in a higher genus Riemann surface. There the bicolourability is no longer automatic and has to be assumed. We thus have two separate countings in non zero genus, that of bicolourable maps and that of general maps. We use a classical method of encoding maps in terms of permutations, on which the constraints of "one-componentness" and of a given genus may be applied. Depending on the orientation issue and on the bicolourability assumption, permutations for a map with n vertices live in S(4n) or in S(2n). In a nutshell, our method reduces to the counting (or listing) of orbits of certain subset of S(4n) (resp. S(2n)) under the action of the centralizer of a certain element of S(4n) (resp. S(2n)). This is achieved either by appealing to a formula by Frobenius or by a direct enumeration of these orbits. Applications to knot theory are briefly mentioned.