The Twist Operator on Maniplexes
Maniplexes are combinatorial objects that generalize, simultaneously, maps on surfaces and abstract polytopes. We are interested on studying highly symmetric maniplexes, particularly those having maximal ‘rotational’ symmetry. This paper introduces an operation on polytopes and maniplexes which, in its simplest form, can be interpreted as twisting the connection between facets. This is first described in detail in dimension 4 and then generalized to higher dimensions. Since the twist on a maniplex preserves all the orientation preserving symmetries of the original maniplex, we apply the operation to reflexible maniplexes, to attack the problem of finding chiral polytopes in higher dimensions.
KeywordsGraph Automorphism group Symmetry Polytope Maniplex Map Flag Transitivity Rotary Reflexible Chiral
We gratefully acknowledge financial support of the PAPIIT-DGAPA, under grant IN107015, and of CONACyT, under grant 166951. The completion of this work was done while the second author was on sabbatical at the Laboratoire d’Informatique de l’École Polytechnique. She thanks LIX and Vincent Pilaud for their hospitality, as well as the program PASPA-DGAPA and the UNAM for the support for this sabbatical stay.
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