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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Archimedean maps of higher genera
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by Ján Karabáš and Roman Nedela PDF
Math. Comp. 81 (2012), 569-583 Request permission

Abstract:

The paper focuses on the classification of vertex-transitive polyhedral maps of genus from $2$ to $4$. These maps naturally generalise the spherical maps associated with the classical Archimedean solids. Our analysis is based on the fact that each Archimedean map on an orientable surface projects onto a one- or a two-vertex quotient map. For a given genus $g\geq 2$ the number of quotients to consider is bounded by a function of $g$. All Archimedean maps of genus $g$ can be reconstructed from these quotients as regular covers with covering transformation group isomorphic to a group $\mathrm {G}$ from a set of $g$-admissible groups. Since the lists of groups acting on surfaces of genus $2,3$ and $4$ are known, the problem can be solved by a computer-aided case-to-case analysis.
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Additional Information
  • Ján Karabáš
  • Affiliation: Science and Research Institute, Matej Bel University, Ďumbierska 1, 974 11 Banská Bystrica, Slovakia
  • Email: karabas@savbb.sk
  • Roman Nedela
  • Affiliation: Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia; Mathematical Institute, Slovak Academy of Sciences, Ďumbierska 1, 974 11 Banská Bystrica, Slovakia
  • MR Author ID: 262779
  • Email: nedela@savbb.sk
  • Received by editor(s): September 14, 2007
  • Received by editor(s) in revised form: November 4, 2010
  • Published electronically: May 13, 2011
  • Additional Notes: Both authors were partially supported by the grants APVV-51-009605 and VEGA 1/0722/08, grants of Slovak Ministry of Education.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 81 (2012), 569-583
  • MSC (2010): Primary 05C30; Secondary 05C10, 05C25
  • DOI: https://doi.org/10.1090/S0025-5718-2011-02502-0
  • MathSciNet review: 2833509