Maps, immersions and permutations

Author:

Coquereaux Robert1,Zuber Jean-Bernard23

Affiliation:

1. Aix Marseille Université, Université de Toulon, CNRS, CPT, UMR 7332, 13288 Marseille, France

2. Sorbonne Universités, UPMC Univ Paris 06, UMR 7589, LPTHE, F-75005 Paris, France

3. CNRS, UMR 7589, LPTHE, F-75005 Paris, France

Abstract

We consider the problem of counting and of listing topologically inequivalent “planar” 4-valent maps with a single component and a given number [Formula: see text] 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 bicolorable and bicolored maps or immersions, where faces are bicolored. Our method extends to immersions of a circle in a higher genus Riemann surface. There the bicolorability is no longer automatic and has to be assumed. We thus have two separate countings in nonzero genus, that of bicolorable 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 bicolorability assumption, permutations for a map with [Formula: see text] vertices live in [Formula: see text] or in [Formula: see text]. In a nutshell, our method reduces to the counting (or listing) of orbits of certain subset of [Formula: see text] (respectively, [Formula: see text]), under the action of the centralizer of a certain element of [Formula: see text] (respectively, [Formula: see text]). 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.

Publisher

World Scientific Pub Co Pte Lt

Subject

Algebra and Number Theory

Reference24 articles.

Cited by 5 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. A complete list of minimal diagrams of an oriented alternating knot;Journal of Knot Theory and Its Ramifications;2021-07

2. A Markov Chain Sampler for Plane Curves;Experimental Mathematics;2019-09-23

3. Asymptotic laws for random knot diagrams;Journal of Physics A: Mathematical and Theoretical;2017-05-09

4. Knot probabilities in random diagrams;Journal of Physics A: Mathematical and Theoretical;2016-09-14

5. A Classification of Spherical Curves Based on Gauss Diagrams;Arnold Mathematical Journal;2016-07-11

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