Mixing chiral polytopes
Gabe Cunningham
Department of Mathematics, Northeastern University, Boston, MA, 02115, USA
DOI: 10.1007/s10801-011-0335-z
Abstract
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.
Pages: 263–277
Keywords: abstract regular polytope; chiral polytope; chiral maps; chirality group
Full Text: PDF
References
Breda D'Azevedo, A., Jones, G.A.: Totally chiral maps and hypermaps of small genus. J. Algebra 322, 3971-3996 (2009) CrossRef Breda D'Azevedo, A., Nedela, R.: Join and intersection of hypermaps. Acta Univ. M. Belli Math. 9, 13-28 (2001) Breda D'Azevedo, A., Jones, G., Nedela, R., Škoviera, M.: Chirality groups of maps and hypermaps. J. Algebr. Comb. 29, 337-355 (2009) CrossRef Breda D'Azevedo, A., Jones, G., Schulte, E.: Constructions of chiral polytopes of small rank. Can. J. Math. $63(6)$, 1254-1283 (2011) CrossRef Breda, A., Breda D'Azevedo, A., Nedela, R.: Chirality group and chirality index of Coxeter chiral maps. Ars Comb. 81, 147-160 (2006) Conder, M., Hubard, I., Pisanski, T.: Constructions for chiral polytopes. J. Lond. Math. Soc. (2) 77, 115-129 (2008) CrossRef Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover, New York (1973) Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, Berlin (1980) Hartley, M.I., Hulpke, A.: Polytopes derived from sporadic simple groups. Contrib. Discret. Math. $5(2)$, 106-118 (2010) Hubard, I., Weiss, A.I.: Self-duality of chiral polytopes. J. Comb. Theory, Ser. A 111, 128-136 (2005) CrossRef Leemans, D., Vauthier, L.: An atlas of abstract regular polytopes for small groups. Aequ. Math. $72(3)$, 313-320 (2006) CrossRef McMullen, P., Schulte, E.: Abstract regular polytopes. In: Encyclopedia of Math. Appl., vol.
92. Cambridge University Press, Cambridge (2002) Orbanic, A.: F-actions and parallel-product decomposition of reflexible maps. J. Algebr. Comb. 26, 507-527 (2007) CrossRef Pellicer, D.: A construction of higher rank chiral polytopes. Discrete Math. 310, 1222-1237 (2010) CrossRef Schulte, E., Weiss, A.I.: Chiral polytopes. In: Gritzmann, P., Sturmfels, B. (eds.) Applied Geometry and Discrete Mathematics (The Klee Festschrift). DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 493-516. Am. Math. Soc./ACM, Providence/New York (1991) Schulte, E., Weiss, A.I.: Free extensions of chiral polytopes. Can. J. Math. 47, 641-654 (1995) CrossRef The GAP Group: GAP-Groups, Algorithms, and Programming, version 4.4.9 (2006). http://www.gap-system.org Wilson, S.E.: Parallel products in groups and maps. J. Algebra 167, 539-546 (1994) CrossRef
92. Cambridge University Press, Cambridge (2002) Orbanic, A.: F-actions and parallel-product decomposition of reflexible maps. J. Algebr. Comb. 26, 507-527 (2007) CrossRef Pellicer, D.: A construction of higher rank chiral polytopes. Discrete Math. 310, 1222-1237 (2010) CrossRef Schulte, E., Weiss, A.I.: Chiral polytopes. In: Gritzmann, P., Sturmfels, B. (eds.) Applied Geometry and Discrete Mathematics (The Klee Festschrift). DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 493-516. Am. Math. Soc./ACM, Providence/New York (1991) Schulte, E., Weiss, A.I.: Free extensions of chiral polytopes. Can. J. Math. 47, 641-654 (1995) CrossRef The GAP Group: GAP-Groups, Algorithms, and Programming, version 4.4.9 (2006). http://www.gap-system.org Wilson, S.E.: Parallel products in groups and maps. J. Algebra 167, 539-546 (1994) CrossRef