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Partial Duality of Hypermaps

Partial Duality of Hypermaps We introduce partial duality of hypermaps, which include the classical Euler–Poincaré duality as a particular case. Combinatorially, hypermaps may be described in one of three ways: as three involutions on the set of flags (bi-rotation system or τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau $$\end{document}-model), or as three permutations on the set of half-edges (rotation system or σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-model in orientable case), or as edge 3-coloured graphs. We express partial duality in each of these models. We give a formula for the genus change under partial duality. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arnold Mathematical Journal Springer Journals

Partial Duality of Hypermaps

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References (49)

Publisher
Springer Journals
Copyright
Copyright © Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2021
ISSN
2199-6792
eISSN
2199-6806
DOI
10.1007/s40598-021-00194-8
Publisher site
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Abstract

We introduce partial duality of hypermaps, which include the classical Euler–Poincaré duality as a particular case. Combinatorially, hypermaps may be described in one of three ways: as three involutions on the set of flags (bi-rotation system or τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau $$\end{document}-model), or as three permutations on the set of half-edges (rotation system or σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-model in orientable case), or as edge 3-coloured graphs. We express partial duality in each of these models. We give a formula for the genus change under partial duality.

Journal

Arnold Mathematical Journal – Springer Journals

Published: Oct 1, 2022

Keywords: Maps; Hypermaps; Partial duality; Permutational models; Rotation system; Bi-rotation system; Edge-coloured graphs; 05C10; 05C65; 57M15; 57Q15

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