Annals of Combinatorics

, Volume 19, Issue 2, pp 243–268 | Cite as

Symmetry Type Graphs of Polytopes and Maniplexes

  • Gabe Cunningham
  • María Del Río-Francos
  • Isabel Hubard
  • Micael Toledo
Article

Abstract

We extend the notion of symmetry type graphs of maps to include maniplexes and (abstract) polytopes, using them to study k-orbit maniplexes (where the automorphism group has k orbits on flags). In particular, we show that there are no fully-transitive k-orbit 3-maniplexes with k > 1 an odd number. We classify 3-orbit maniplexes and determine all face transitivities for 3- and 4-orbit maniplexes. Moreover, we give generators of the automorphism group of a maniplex, given its symmetry type graph. Finally, we extend these notions to oriented maniplexes, and we provide a classification of oriented 2-orbit maniplexes and a generating set for their orientation-preserving automorphism group.

Mathematics Subject Classification

52B15 05C25 51M20 

Keywords

abstract polytope regular graph edge-colouring maniplex 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brinkmann G., Van Cleemput N., Pisanski T.: Generation of various classes of trivalent graphs. Theoret. Comput. Sci. 502, 16–29 (2013)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Coxeter, H.S.M, Moser, W.O.J.: Generators and Relations for Discrete Groups. Springer-Verlag, Berlin-Göttigen-Heidelberg (1957)Google Scholar
  3. 3.
    Del Río-Francos, M.: Truncation symmetry type graphs. Submitted.Google Scholar
  4. 4.
    Delgado Friedrichs O., Huson D.H.: Tiling space by Platonic solids. I. Discrete Comput. Geom. 21(2), 299–315 (1999)MATHMathSciNetGoogle Scholar
  5. 5.
    Dress A.W.M.: Presentations of discrete groups, acting on simply connected manifolds, in terms of parametrized systems of Coxeter matrices—a systematic approach. Adv. Math. 63(2), 196–212 (1987)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Dress A.W.M., Huson D.H.: On tilings of the plane. Geom. Dedicata 24(3), 295–310 (1987)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Hartley M.I.: All polytopes are quotients, and isomorphic polytopes are quotients by conjugate subgroups. Discrete Comput. Geom. 21(2), 289–298 (1999)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Holt, D.F., Eick, B., O’Brien, E.A.: Handbook of Computational Group Theory. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL (2005)Google Scholar
  9. 9.
    Hubard I.: Two-orbit polyhedra from groups. European J. Combin. 31(3), 943–960 (2010)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hubard, I.A.: From Geometry to Groups and Back: The Study of Highly Symmetric Polytopes. Ph.D. Thesis, York University, Toronto (2007)Google Scholar
  11. 11.
    Hubard, I., Del Río-Francos, M., Orbanic, A., Pisanski, T.: Medial symmetry type graphs. Electron. J. Combin. 20(3), #P29 (2013)Google Scholar
  12. 12.
    Hubard I., Orbanić A., Weiss A.I.: Monodromy groups and self-invariance. Canad. J. Math. 61(6), 1300–1324 (2009)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Huson D.H.: The generation and classification of tile-k-transitive tilings of Euclidean plane, the sphere and the hyperbolic plane. Geom. Dedicata 47(3), 269–296 (1993)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Kocič J.: Symmetry-type graphs of Platonic and Archimedean solids. Math. Commun. 16(2), 491–507 (2011)MathSciNetGoogle Scholar
  15. 15.
    McMullen P.: Regular polytopes of nearly full rank: addendum. Discrete Comput. Geom. 49(3), 703–705 (2013)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    McMullen, P., Schulte, E.: Abstract Regular Polytopes. Encyclopedia Math. Appl. 92. Cambridge University Press, Cambridge (2002)Google Scholar
  17. 17.
    Orbanić A., Pellicer D., Weiss A.I.: Map operation and k-orbit maps. J. Combin. Theory Ser. A 117(4), 411–429 (2010)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Pisanski, T.: Personal communication.Google Scholar
  19. 19.
    Schulte, E., Weiss, A.I.: Chiral polytopes. In: Gritzmann, P., Sturmfels, B. (eds.) Applied Geometry and Discrete Mathematics, pp. 493–516. Amer. Math. Soc., Providence, RI (1991)Google Scholar
  20. 20.
    Širáň J., Tucker T.W., Watkins M.E.: Realizing finite edge-transitive orientable maps. J. Graph Theory 37(1), 1–34 (2001)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Wilson S.: Maniplexes: part 1: maps, polytopes, symmetry and operators. Symmetry 4(2), 265–275 (2012)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Gabe Cunningham
    • 1
  • María Del Río-Francos
    • 2
  • Isabel Hubard
    • 3
    • 4
  • Micael Toledo
    • 3
    • 4
  1. 1.University of Massachusetts BostonBostonUSA
  2. 2.Institute of Mathematics Physics and MechanicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Instituto de Matemáticas, Universidad Nacional Autónoma de MéxicoCuernavacaMéxico
  4. 4.CuernavacaMéxico

Personalised recommendations