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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Archimedean maps of higher genera


Authors: Ján Karabáš and Roman Nedela
Journal: Math. Comp. 81 (2012), 569-583
MSC (2010): Primary 05C30; Secondary 05C10, 05C25
Published electronically: May 13, 2011
MathSciNet review: 2833509
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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
Email: nedela@savbb.sk

DOI: http://dx.doi.org/10.1090/S0025-5718-2011-02502-0
PII: S 0025-5718(2011)02502-0
Keywords: Polyhedron, Archimedean solid, map, surface, group, graph embedding
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.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.