Stratified operations on maniplexes
Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 267-287.

There is an increasingly extensive literature on the problem of describing the connection (monodromy) groups and automorphism groups of families of polytopes and maniplexes that are not regular or reflexible. Many such polytopes and maniplexes arise as the result of constructions such as truncations and products. Here we show that for a wide variety of these constructions, the connection group of the output can be described in a nice way in terms of the connection group of the input. We call such operations stratified. Moreover, we show that, if F is a maniplex operation in one of two broad subclasses of stratified operations, and if is the smallest reflexible cover of some maniplex , then the connection group of F() is equal to the connection group of F(). In particular, we show that this is true for truncations and medials of maps, for products of polytopes (including pyramids and prisms over polytopes), and for the mix of maniplexes. As an application, we determine the smallest reflexible covers of the pyramids over the equivelar toroidal maps.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.208
Classification: 52B15, 05E18, 52B05
Keywords: Polytope, maniplex, connection group, monodromy group, truncation, medial, pyramid, prism

Cunningham, Gabe 1; Pellicer, Daniel 2; Williams, Gordon 3

1 University of Massachusetts Boston Department of Mathematics Boston, Massachusetts, USA
2 Centro de Ciencias Matemáticas, UNAM Morelia, Mexico
3 University of Alaska Fairbanks Department of Mathematics Fairbanks, Alaska, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Cunningham, Gabe; Pellicer, Daniel; Williams, Gordon. Stratified operations on maniplexes. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 267-287. doi : 10.5802/alco.208. https://alco.centre-mersenne.org/articles/10.5802/alco.208/

[1] Berman, Leah Wrenn; Mixer, Mark; Monson, Barry; Oliveros, Deborah; Williams, Gordon The monodromy group of the n-pyramid, Discrete Math., Volume 320 (2014), pp. 55-63 | DOI | MR | Zbl

[2] Berman, Leah Wrenn; Monson, Barry; Oliveros, Deborah; Williams, Gordon The monodromy group of a truncated simplex, J. Algebraic Combin., Volume 42 (2015) no. 3, pp. 745-761 | DOI | MR | Zbl

[3] Brehm, Ulrich; Kühnel, Wolfgang Equivelar maps on the torus, European J. Combin., Volume 29 (2008) no. 8, pp. 1843-1861 | DOI | MR | Zbl

[4] Conder, Marston The smallest regular polytopes of given rank, Adv. Math., Volume 236 (2013), pp. 92-110 | DOI | MR | Zbl

[5] Cunningham, Gabe Self-dual, self-Petrie covers of regular polyhedra, Symmetry, Volume 4 (2012) no. 1, pp. 208-218 | DOI | MR | Zbl

[6] Cunningham, Gabe Flat extensions of abstract polytopes, Art Discrete Appl. Math., Volume 4 (2021) no. 3, Paper no. 3.06, 14 pages | DOI | MR

[7] Cunningham, Gabe; Del Río-Francos, María; Hubard, Isabel; Toledo, Micael Symmetry type graphs of polytopes and maniplexes, Ann. Comb., Volume 19 (2015) no. 2, pp. 243-268 | DOI | MR | Zbl

[8] Cunningham, Gabe; Pellicer, Daniel Open problems on k-orbit polytopes, Discrete Math., Volume 341 (2018) no. 6, pp. 1645-1661 | DOI | MR | Zbl

[9] Danzer, Ludwig Regular incidence-complexes and dimensionally unbounded sequences of such. I, Convexity and graph theory (Jerusalem, 1981) (North-Holland Math. Stud.), Volume 87, North-Holland, Amsterdam, 1984, pp. 115-127 | DOI | MR | Zbl

[10] Douglas, Ian; Hubard, Isabel; Pellicer, Daniel; Wilson, Steve The twist operator on maniplexes, Discrete geometry and symmetry (Springer Proc. Math. Stat.), Volume 234, Springer, Cham, 2018, pp. 127-145 | DOI | MR | Zbl

[11] Drach, Kostiantyn; Mixer, Mark Minimal covers of equivelar toroidal maps, Ars Math. Contemp., Volume 9 (2015) no. 1, pp. 77-91 | DOI | MR | Zbl

[12] Fernandes, Maria Elisa; Leemans, Dimitri; Mixer, Mark Polytopes of high rank for the alternating groups, J. Combin. Theory Ser. A, Volume 119 (2012) no. 1, pp. 42-56 | DOI | MR | Zbl

[13] GAP – Groups, Algorithms, and Programming, Version 4.4.12 (2008) (http://www.gap-system.org)

[14] Garza-Vargas, Jorge; Hubard, Isabel Polytopality of maniplexes, Discrete Math., Volume 341 (2018) no. 7, pp. 2068-2079 | DOI | MR | Zbl

[15] Gleason, Ian; Hubard, Isabel Products of abstract polytopes, J. Combin. Theory Ser. A, Volume 157 (2018), pp. 287-320 | DOI | MR | Zbl

[16] Grünbaum, Branko Convex polytopes, Graduate Texts in Mathematics, 221, Springer-Verlag, New York, 2003, xvi+468 pages (Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler) | DOI | MR | Zbl

[17] Hartley, Michael I.; Hulpke, Alexander Polytopes derived from sporadic simple groups, Contrib. Discrete Math., Volume 5 (2010) no. 2, pp. 106-118 | MR | Zbl

[18] Hartley, Michael I.; Pellicer, Daniel; Williams, Gordon Minimal covers of the prisms and antiprisms, Discrete Math., Volume 312 (2012) no. 20, pp. 3046-3058 | DOI | MR | Zbl

[19] Hartley, Michael I.; Williams, Gordon Representing the sporadic Archimedean polyhedra as abstract polytopes, Discrete Math., Volume 310 (2010) no. 12, pp. 1835-1844 | DOI | MR | Zbl

[20] Helfand, Ilanit Constructions of k-orbit Abstract Polytopes, ProQuest LLC, Ann Arbor, MI, 2013, 147 pages Thesis (Ph.D.)–Northeastern University | MR

[21] Hubard, Isabel Two-orbit polyhedra from groups, European J. Combin., Volume 31 (2010) no. 3, pp. 943-960 | DOI | MR | Zbl

[22] Hubard, Isabel; del Río Francos, María; Orbanić, Alen; Pisanski, Tomaž Medial symmetry type graphs, Electron. J. Combin., Volume 20 (2013) no. 3, Paper no. 29, 28 pages | DOI | MR | Zbl

[23] Hubard, Isabel; Orbanić, Alen; Pellicer, Daniel; Weiss, Asia Ivić Symmetries of equivelar 4-toroids, Discrete Comput. Geom., Volume 48 (2012) no. 4, pp. 1110-1136 | DOI | MR | Zbl

[24] Hubard, Isabel A. From geometry to groups and back: The study of highly symmetric polytopes, ProQuest LLC, Ann Arbor, MI, 2007, 168 pages Thesis (Ph.D.)–York University (Canada) | MR

[25] Jones, Gareth A.; Poulton, Andrew Maps admitting trialities but not dualities, European J. Combin., Volume 31 (2010) no. 7, pp. 1805-1818 | DOI | MR | Zbl

[26] Koike, Hiroki; Pellicer, Daniel; Raggi, Miguel; Wilson, Steve Flag bicolorings, pseudo-orientations, and double covers of maps, Electron. J. Combin., Volume 24 (2017) no. 1, Paper no. 1.3, 23 pages | DOI | MR | Zbl

[27] McMullen, Peter; Schulte, Egon Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, 92, Cambridge University Press, Cambridge, 2002, xiv+551 pages | DOI | MR | Zbl

[28] Mixer, Mark; Pellicer, Daniel; Williams, Gordon Minimal covers of the Archimedean tilings, part II, Electron. J. Combin., Volume 20 (2013) no. 2, Paper no. 20, 19 pages | DOI | MR | Zbl

[29] Monson, Barry; Pellicer, Daniel; Williams, Gordon Mixing and monodromy of abstract polytopes, Trans. Amer. Math. Soc., Volume 366 (2014) no. 5, pp. 2651-2681 | DOI | MR | Zbl

[30] Orbanić, Alen F-actions and parallel-product decomposition of reflexible maps, J. Algebraic Combin., Volume 26 (2007) no. 4, pp. 507-527 | DOI | MR | Zbl

[31] Orbanić, Alen; Pellicer, Daniel; Weiss, Asia Ivić Map operations and k-orbit maps, J. Combin. Theory Ser. A, Volume 117 (2010) no. 4, pp. 411-429 | DOI | MR | Zbl

[32] Pellicer, Daniel Cleaved abstract polytopes, Combinatorica, Volume 38 (2018) no. 3, pp. 709-737 | DOI | MR | Zbl

[33] Pellicer, Daniel; Williams, Gordon Minimal covers of the Archimedean tilings, Part 1, Electron. J. Combin., Volume 19 (2012) no. 3, Paper no. 6, 37 pages | MR | Zbl

[34] Pellicer, Daniel; Williams, Gordon Pyramids over regular 3-tori, SIAM J. Discrete Math., Volume 32 (2018) no. 1, pp. 249-265 | DOI | MR | Zbl

[35] Richter, R. Bruce; Širáň, Jozef; Wang, Yan Self-dual and self-Petrie-dual regular maps, J. Graph Theory, Volume 69 (2012) no. 2, pp. 152-159 | DOI | MR | Zbl

[36] Schulte, Egon Amalgamation of regular incidence-polytopes, Proc. London Math. Soc. (3), Volume 56 (1988) no. 2, pp. 303-328 | DOI | MR | Zbl

[37] Vince, Andrew Regular combinatorial maps, J. Combin. Theory Ser. B, Volume 35 (1983) no. 3, pp. 256-277 | DOI | MR | Zbl

[38] Wilson, Stephen E. Parallel products in groups and maps, J. Algebra, Volume 167 (1994) no. 3, pp. 539-546 | DOI | MR | Zbl

[39] Wilson, Steve Maniplexes: Part 1: maps, polytopes, symmetry and operators, Symmetry, Volume 4 (2012) no. 2, pp. 265-275 | DOI | MR | Zbl

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