publicações

Journal Articles

2025

1. J. Combin. Theory Ser. A

   Cayley extensions of maniplexes and polytopes

   Gabe Cunningham, Elías Mochán, e Antonio Montero

   Journal of Combinatorial Theory, Series A, May 2025

   Abs DOI arXiv Bib MathSciNet

   

   A map on a surface whose automorphism group has a subgroup acting regularly on its vertices is called a Cayley map. Here we generalize that notion to maniplexes and polytopes. We define M to be a Cayley extension of K if the facets of M are isomorphic to K and if some subgroup of the automorphism group of M acts regularly on the facets of M. We show that many natural extensions in the literature on maniplexes and polytopes are in fact Cayley extensions. We also describe several universal Cayley extensions. Finally, we examine the automorphism group and symmetry type graph of Cayley extensions.

   @article{CunMocMon2025_CayleyExtensionsManiplexes,
     author = {Cunningham, Gabe and Mochán, Elías and Montero, Antonio},
     journal = {Journal of Combinatorial Theory, Series A},
     title = {Cayley extensions of maniplexes and polytopes},
     year = {2025},
     issn = {0097-3165},
     month = may,
     pages = {106000},
     volume = {212},
     doi = {10.1016/j.jcta.2024.106000},
     keywords = {Maniplexes, Maps, Abstract polytopes},
     mrnumber = {4841625},
     url = {https://www.sciencedirect.com/science/article/pii/S0097316524001390},
     urldate = {2024-12-18}
   }

2. Combinatorica

   Chiral Extensions of Regular Toroids

   Antonio Montero e Micael Toledo

   Combinatorica, May 2025

   Abs DOI arXiv Bib MathSciNet

   

   Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational symmetry but do not admit reflections. In this paper we build chiral polytopes whose facets (maximal faces) are isomorphic to a prescribed regular cubic tessellation of the n-dimensional torus \(n ≥2\). As a consequence, we prove that for every \(d ≥3\)there exist infinitely many chiral d-polytopes.

   @article{MonTol2025_ChiralExtensionsRegular,
     author = {Montero, Antonio and Toledo, Micael},
     journal = {Combinatorica},
     title = {Chiral {Extensions} of {Regular} {Toroids}},
     year = {2025},
     issn = {1439-6912},
     month = may,
     number = {1},
     pages = {5},
     volume = {45},
     doi = {10.1007/s00493-024-00132-0},
     keywords = {Abstract polytopes, Chiral polytopes, Regular toroids, Primary: 52B15, Secondary: 52C22, 05E18},
     language = {en},
     mrnumber = {4844340},
     url = {https://doi.org/10.1007/s00493-024-00132-0},
     urldate = {2024-12-29}
   }

3. European J. Combin.

   Symmetries of voltage operations on polytopes, maps and maniplexes

   Isabel Hubard, Elías Mochán, e Antonio Montero

   European Journal of Combinatorics, May 2025

   Abs DOI arXiv Bib MathSciNet

   

   Voltage operations extend traditional geometric and combinatorial operations (such as medial, truncation, prism, and pyramid over a polytope) to operations on maniplexes, maps, polytopes, and hypertopes. In classical operations, the symmetries of the original object remain in the resulting one, but sometimes additional symmetries are created; the same situation arises with voltage operations. We characterise the automorphisms of the new object that are derived from the original one and use this to bound the number of flag orbits (under the action of its automorphism group) of the new object in terms of the original one. The conditions under which the automorphism group of the original object is the same as the automorphism group of the resulting object are given. We also look at the cases where there is additional symmetry which can be accurately described due to the symmetries of the operation itself.

   @article{HubMocMon2025_SymmetriesVoltageOperations,
     author = {Hubard, Isabel and Mochán, Elías and Montero, Antonio},
     journal = {European Journal of Combinatorics},
     title = {Symmetries of voltage operations on polytopes, maps and maniplexes},
     year = {2025},
     issn = {0195-6698},
     month = may,
     pages = {104132},
     volume = {126},
     doi = {10.1016/j.ejc.2025.104132},
     mrnumber = {4860806},
     url = {https://www.sciencedirect.com/science/article/pii/S0195669825000149},
     urldate = {2025-02-08}
   }

4. J. Combin. Theory Ser. A

   Vertex-transitive graphs with small motion and transitive permutation groups with small minimal degree

   Antonio Montero e Primož Potočnik

   Journal of Combinatorial Theory, Series A, Nov 2025

   Abs DOI arXiv Bib MathSciNet

   

   The motion of a graph is the minimum number of vertices that are moved by a non-trivial automorphism. Equivalently, it can be defined as the minimal degree of its automorphism group (as a permutation group on the vertices). In this paper, we develop some results on permutation groups (primitive and imprimitive) with small minimal degree. As a consequence of such results, we classify vertex-transitive graphs whose motion is 4 or a prime number.

   @article{MonPot2025_VertexTransitiveGraphs,
     author = {Montero, Antonio and Potočnik, Primož},
     journal = {Journal of Combinatorial Theory, Series A},
     title = {Vertex-transitive graphs with small motion and transitive permutation groups with small minimal degree},
     year = {2025},
     issn = {0097-3165},
     month = nov,
     pages = {106065},
     volume = {216},
     doi = {10.1016/j.jcta.2025.106065},
     keywords = {Fixity, Motion, Minimal degree, Graphs, Permutation groups},
     mrnumber = {4910474},
     url = {https://www.sciencedirect.com/science/article/pii/S0097316525000603},
     urldate = {2025-05-22}
   }

2023

1. Combinatorica

   Voltage Operations on Maniplexes, Polytopes and Maps

   Isabel Hubard, Elías Mochán, e Antonio Montero

   Combinatorica, May 2023

   Abs DOI arXiv Bib MathSciNet

   

   Classical geometric and topological operations on polyhedra, maps and polytopes often give rise to structures with the same symmetry group as the original one, but with more flags. In this paper we introduce the notion of voltage operations on maniplexes, as a way to unify the study of such operations and generalize them to other geometric and combinatorial structures such as abstract polytopes, hypermaps, maniplexes or hypertopes. This can be done since our technique provides a way to study classical operations in a graph theoretic setting, and thus to apply a voltage operation one only needs that the combinatorial structure in hand can be understood as an n-valent properly n-edge colored graph. For example, in the case of abstract polytopes, the partial order can be encoded into the so-called flag graph of the polytope and the voltage operation is therefore applied to such flag graph to then be recovered as a partial order. We focus on studying the interactions between voltage operations and the symmetries of the operated object, and show that these operations can be potentially used to build maniplexes with prescribed symmetry type graphs. Moreover, a complete characterization of when an operation can be seen as a voltage operation is given.

   @article{HubMocMon2023_VoltageOperationsManiplexes,
     author = {Hubard, Isabel and Mochán, Elías and Montero, Antonio},
     journal = {Combinatorica},
     title = {Voltage {Operations} on {Maniplexes}, {Polytopes} and {Maps}},
     year = {2023},
     issn = {1439-6912},
     month = may,
     doi = {10.1007/s00493-023-00018-7},
     keywords = {Abstract polytopes, Map operations, Operations, Maniplexes, Voltage graphs, 52B15, 05E18, 52B05},
     language = {en},
     mrnumber = {4627315},
     url = {https://doi.org/10.1007/s00493-023-00018-7},
     urldate = {2023-05-25}
   }

2021

1. Discrete Comput. Geom.

   Equivelar Toroids with Few Flag-Orbits

   José Collins e Antonio Montero

   Discrete & Computational Geometry, Jan 2021

   Abs DOI arXiv Bib MathSciNet

   

   An (n+1)-toroid is a quotient of a tessellation of the n-dimensional Euclidean space with a lattice group. Toroids are generalisations of maps on the torus to higher dimensions and also provide examples of abstract polytopes. Equivelar toroids are those that are induced by regular tessellations. In this paper we present a classification of equivelar (n+1)-toroids with at most n flag-orbits; in particular, we discuss a classification of 2-orbit toroids of arbitrary dimension.

   @article{ColMon2021_EquivelarToroidsFew,
     author = {Collins, José and Montero, Antonio},
     journal = {Discrete \& Computational Geometry},
     title = {Equivelar {Toroids} with {Few} {Flag}-{Orbits}},
     year = {2021},
     issn = {1432-0444},
     month = jan,
     number = {2},
     pages = {305--330},
     volume = {65},
     doi = {10.1007/s00454-020-00230-y},
     keywords = {research},
     language = {en},
     mrnumber = {4212966},
     url = {https://doi.org/10.1007/s00454-020-00230-y},
     urldate = {2021-02-18}
   }

2. Discrete Math.

   On the Schläfli symbol of chiral extensions of polytopes

   Antonio Montero

   Discrete Mathematics, Nov 2021

   Abs DOI arXiv Bib MathSciNet

   

   Given an abstract n-polytope K, an abstract (n+1)-polytope P is an extension of K if all the facets of P are isomorphic to K. A chiral polytope is a polytope with maximal rotational symmetry that does not admit any reflections. If P is a chiral extension of K, then all but the last entry of the Schläfli symbol of P are determined. In this paper we introduce some constructions of chiral extensions P of certain chiral polytopes in such a way that the last entry of the Schläfli symbol of P is arbitrarily large.

   @article{Mon2021_SchlaefliSymbolChiral,
     author = {Montero, Antonio},
     journal = {Discrete Mathematics},
     title = {On the {Schläfli} symbol of chiral extensions of polytopes},
     year = {2021},
     issn = {0012-365X},
     month = nov,
     number = {11},
     pages = {112507},
     volume = {344},
     doi = {10.1016/j.disc.2021.112507},
     keywords = {Abstract polytopes, Chiral polytopes, Schläfli symbol, research},
     language = {en},
     mrnumber = {4298160},
     url = {https://www.sciencedirect.com/science/article/pii/S0012365X2100220X},
     urldate = {2021-09-17}
   }

3. J. Algebraic Combin.

   Proper locally spherical hypertopes of hyperbolic type

   Antonio Montero e Asia Ivić Weiss

   Journal of Algebraic Combinatorics, Oct 2021

   Abs DOI arXiv Bib MathSciNet

   

   Given any irreducible Coxeter group C of hyperbolic type with nonlinear diagram and rank at least 4, whose maximal parabolic subgroups are finite, we construct an infinite family of locally spherical regular hypertopes of hyperbolic type whose Coxeter diagram is the same as that of C.

   @article{MonWei2021_ProperLocallySpherical,
     author = {Montero, Antonio and Weiss, Asia Ivić},
     journal = {Journal of Algebraic Combinatorics},
     title = {Proper locally spherical hypertopes of hyperbolic type},
     year = {2021},
     issn = {1572-9192},
     month = oct,
     doi = {10.1007/s10801-021-01054-6},
     keywords = {research},
     language = {en},
     mrnumber = {4390589},
     url = {https://doi.org/10.1007/s10801-021-01054-6},
     urldate = {2021-10-25}
   }

4. Art Discrete Appl. Math.

   Locally spherical hypertopes from generalised cubes

   Antonio Montero e Asia Ivić Weiss

   Art Discrete Appl. Math., Ago 2021

   Abs DOI arXiv Bib MathSciNet

   

   We show that every non-degenerate regular polytope can be used to construct a thin, residually-connected, chamber-transitive incidence geometry, i.e. a regular hypertope. These hypertopes are related to the semi-regular polyotopes with a tail-triangle Coxeter diagram constructed by Monson and Schulte. We discuss several interesting examples derived when this construction is applied to generalised cubes. In particular, we produce an example of a rank 5 finite locally spherical proper hypertope of hyperbolic type. No such examples were previously known.

   @article{MonWei2021_LocallySphericalHypertopes,
     author = {Montero, Antonio and Weiss, Asia Ivić},
     journal = {Art Discrete Appl. Math.},
     title = {Locally spherical hypertopes from generalised cubes},
     year = {2021},
     month = {Ago},
     number = {3},
     pages = {Paper No. 3.07, 12},
     volume = {4},
     doi = {10.26493/2590-9770.1354.b40},
     fjournal = {The Art of Discrete and Applied Mathematics},
     keywords = {research},
     mrclass = {52B15 (51E24 51G05)},
     mrnumber = {4312691},
     mrreviewer = {Frieder Ladisch},
     url = {https://doi.org/10.26493/2590-9770.1354.b40}
   }

2018

1. Adv. Geom.

   Regular polyhedra in the 3-torus

   Antonio Montero

   Advances in Geometry, Oct 2018

   Abs DOI arXiv Bib MathSciNet

   

   In this paper we discuss the classification rank 3 lattices preserved by finite orthogonal groups and derive from it the classification of regular polyhedra in the 3-dimensional torus. This classification is closely related to the classification of regular polyhedra in the 3-space.

   @article{Mon2018_RegularPolyhedra3,
     author = {Montero, Antonio},
     journal = {Advances in Geometry},
     title = {Regular polyhedra in the 3-torus},
     year = {2018},
     issn = {1615-715X},
     month = oct,
     number = {4},
     pages = {431--450},
     volume = {18},
     doi = {10.1515/advgeom-2018-0017},
     keywords = {52B15 (51M20 52B10), research},
     mrnumber = {3871407}
   }

Theses

2019

1. Ph.D.

   Chiral extensions of toroids

   Antonio Montero

   UNAM-UMSNH, 2019

   Bib HTML PDF

   @phdthesis{Mon2019_ChiralExtensionsToroids_PhDThesis,
     author = {Montero, Antonio},
     school = {UNAM-UMSNH},
     title = {Chiral extensions of toroids},
     year = {2019},
     type = {phdthesis},
     keywords = {thesis},
   }

2015

1. M.Sc.

   Realizaciones regulares de poliedros en el 3-toro (Realizations of regular polyhedra in the 3-torus)

   Antonio Montero

   UNAM-UMSNH, 2015

   Bib HTML PDF

   @mastersthesis{Mon2015_RealizacionesRegularesDe_MScThesis,
     author = {Montero, Antonio},
     school = {UNAM-UMSNH},
     title = {Realizaciones regulares de poliedros en el $3$-toro ({R}ealizations of regular polyhedra in the $3$-torus)},
     year = {2015},
     type = {mastersthesis},
     keywords = {thesis},
   }

2013

1. B.Sc.

   Poliedros regulares en el 3-toro (Regular polyhedra in the 3-torus)

   Antonio Montero

   UMSNH, 2013

   Bib HTML PDF

   @mastersthesis{Mon2013_PoliedrosRegularesEn_BScThesis,
     author = {Montero, Antonio},
     school = {UMSNH},
     title = {Poliedros regulares en el $3$-toro ({R}egular polyhedra in the $3$-torus)},
     year = {2013},
     type = {thesis},
     keywords = {thesis},
   }

Miscellaneous

2016

1. Outreach

   Simetrías: Las matemáticas detrás de la belleza (Symmetry: the mathematics behind beauty)

   Antonio Montero

   2016

   HTML
2. Outreach

   Simetrías de polígonos, poliedros y otras policosas (Symmetries of polygons, polyhedra and other poly-things)

   Antonio Montero

   2016

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