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Chirality groups of maps and hypermaps

Chirality groups of maps and hypermaps
Chirality groups of maps and hypermaps
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.
0925-9899
337-355
Breda d'Azevedo, Antonio
ea38b7ac-138c-4cc7-97fb-fd271d4243cf
Jones, Gareth
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
Nedela, Roman
229c07a1-a5fa-477c-88e4-e591a65bb644
Skoviera, Martin
6d47da7c-1e1b-4e74-82bb-39d52c1ef63a
Breda d'Azevedo, Antonio
ea38b7ac-138c-4cc7-97fb-fd271d4243cf
Jones, Gareth
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
Nedela, Roman
229c07a1-a5fa-477c-88e4-e591a65bb644
Skoviera, Martin
6d47da7c-1e1b-4e74-82bb-39d52c1ef63a

Breda d'Azevedo, Antonio, Jones, Gareth, Nedela, Roman and Skoviera, Martin (2009) Chirality groups of maps and hypermaps Journal of Algebraic Combinatorics, 29, (3), pp. 337-355. (doi:10.1007/s10801-008-0138-z).

Record type: Article

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.

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Published date: May 2009

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Local EPrints ID: 156833
URI: http://eprints.soton.ac.uk/id/eprint/156833
ISSN: 0925-9899
PURE UUID: 40e240c4-4f20-4203-98b8-7766ec4d91e0

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Date deposited: 02 Jun 2010 12:02
Last modified: 18 Jul 2017 12:41

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Author: Antonio Breda d'Azevedo
Author: Gareth Jones
Author: Roman Nedela
Author: Martin Skoviera

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