4 research outputs found
Constructing Self-Dual Chiral Polytopes
An abstract polytope is chiral if its automorphism group has two orbits on
the flags, such that adjacent flags belong to distinct orbits. There are still
few examples of chiral polytopes, and few constructions that can create chiral
polytopes with specified properties. In this paper, we show how to build
self-dual chiral polytopes using the mixing construction for polytopes.Comment: 16 page
Mixing Chiral Polytopes
An abstract polytope of rank n is said to be chiral if its automorphism group
has two orbits on the flags, such that adjacent flags belong to distinct
orbits. Examples of chiral polytopes have been difficult to find. A "mixing"
construction lets us combine polytopes to build new regular and chiral
polytopes. By using the chirality group of a polytope, we are able to give
simple criteria for when the mix of two polytopes is chiral
Bicontactual hypermaps
Doutoramento em MatemáticaEsta tese dedica-se ao estudo de hipermapas regulares bicontactuais, hipermapas com a propriedade que cada hiperface contacta só com outras duas hiperfaces. Nos anos 70, S. Wilson classificou os mapas bicontactuais e, em 2003, Wilson e Breda d’Azevedo classificaram os hipermapas bicontactuais no caso não-orientável. Quando esta propriedade é transferida para
hipermapas origina trĂŞs tipos de bicontactualidade, atendendo ao modo como as duas hiperfaces aparecem Ă volta de uma hiperface fixa: edge-twin, vertextwin and alternate (dois deles sĂŁo o dual um do outro).
Um hipermapa topolĂłgico Ă© um mergulho celular de um grafo conexo trivalente numa superfĂcie compacta e conexa tal que as cĂ©lulas sĂŁo 3-coloridas. Ou de maneira mais simples, um hipermapa pode ser visto como um mapa bipartido.
Um hipermapa orientado regular é um triplo ordenado consistindo num conjunto finito e dois geradores, que são permutações (involuções) do conjunto tal que o grupo gerado por eles, chamado o grupo de monodromia, actua regularmente no conjunto.
Nesta tese, damos uma classificação de todos os hipermapas orientados regulares bicontactuais e, para completar, reclassificamos, usando o nosso
método algébrico, os hipermapas não-orientáveis bicontactuais.This thesis is devoted to the study of bicontactual regular hypermaps, hypermaps with the property that each hyperface meets only two others. In the
seventies, S. Wilson classified the bicontactual maps and, in 2003, Wilson and Breda d’Azevedo classified the bicontactual non-orientable hypermaps. When this property is transferred for hypermaps it gives rise to three types of
bicontactuality, according as the two hyperfaces appear around a fixed hyperface: edge-twin, vertex-twin and alternate (two of which are dual of each other).
A topological hypermap is a cellular embedding of a connected trivalent graph into a compact and connected surface such that the cells are 3-colored. Or simply, a hypermap can be seen as a bipartite map.
A regular oriented-hypermap is an ordered triple, consisting of a finite set and two generators, which are permutations of the set, such that the group
generate by them, called monodromy group, acts regularly on the set.
In this thesis, we give a classification of all bicontactual regular orientedhypermaps and, for completion, we reclassify, using our algebraic method, the bicontactual non-orientable hypermaps