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The Twist Operator on Maniplexes

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Discrete Geometry and Symmetry (GSC 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 234))

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. 17 E Pennington St, Tucson, AZ, 85701, USA

    Ian Douglas

  2. Instituto de Matemáticas, Universidad Nacional Autónoma de México Circuito Exterior, C.U., Coyoacán, 04510, Mexico D.F., Mexico

    Isabel Hubard

  3. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Antigua Carretera a Pátzcuaro 8701, Ex Hacienda de San José de la Huerta, 58089, Morelia, Michoacán, Mexico

    Daniel Pellicer

  4. Department of Mathematics and Statistics, Northern Arizona University, Box 5717, Flagstaff, AZ, 86011, USA

    Steve Wilson

Authors
  1. Ian Douglas
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  2. Isabel Hubard
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  3. Daniel Pellicer
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  4. Steve Wilson
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Corresponding author

Correspondence to Steve Wilson .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Auckland, Auckland, New Zealand

    Marston D. E. Conder

  2. Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada

    Antoine Deza

  3. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

    Asia Ivić Weiss

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Douglas, I., Hubard, I., Pellicer, D., Wilson, S. (2018). The Twist Operator on Maniplexes. In: Conder, M., Deza, A., Weiss, A. (eds) Discrete Geometry and Symmetry. GSC 2015. Springer Proceedings in Mathematics & Statistics, vol 234. Springer, Cham. https://doi.org/10.1007/978-3-319-78434-2_7

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