Chirality groups of maps and hypermaps

  • Antonio Breda D’Azevedo
  • Gareth Jones
  • Roman Nedela
  • Martin Škoviera
Article

DOI: 10.1007/s10801-008-0138-z

Cite this article as:
Breda D’Azevedo, A., Jones, G., Nedela, R. et al. J Algebr Comb (2009) 29: 337. doi:10.1007/s10801-008-0138-z

Abstract

Although the phenomenon of chirality appears in many investigations of maps and hypermaps, no detailed study of chirality seems to have been carried out. Chirality of maps and hypermaps is not merely a binary invariant but can be quantified by two new invariants—the chirality group and the chirality index, the latter being the size of the chirality group. A detailed investigation of the chirality groups of orientably regular maps and hypermaps will be the main objective of this paper. The most extreme type of chirality arises when the chirality group coincides with the monodromy group. Such hypermaps are called totally chiral. Examples of these are constructed by considering appropriate “asymmetric” pairs of generators of certain non-abelian simple groups. We also show that every finite abelian group is the chirality group of some hypermap, whereas many non-abelian groups, including symmetric and dihedral groups, cannot arise as chirality groups.

Keywords

Map Hypermap Chiral Asymmetric Chirality group Chirality index 

Mathematics Subject Classification (2000)

05C10 05C25 20B25 

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Antonio Breda D’Azevedo
    • 1
  • Gareth Jones
    • 2
  • Roman Nedela
    • 3
  • Martin Škoviera
    • 4
  1. 1.Departamento de MatematicaUniversidade de AveiroAveiroPortugal
  2. 2.School of MathematicsUniversity of SouthamptonSouthamptonUK
  3. 3.Department of MathematicsMatej Bel UniversityBanská BystricaSlovakia
  4. 4.Department of InformaticsComenius UniversityBratislavaSlovakia

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