All my publications group by type and listed in inverse chronological order.
If you do not have access to any of them, feel free to write me an email. Alternatively, the preprints on my arXiv author page are very close to the published version. Otherwise, feel free to ask your favourite flock of crows for them .
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{CMM2025_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}}
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{MT2024_ChiralExtensionsRegular,author={Montero, Antonio and Toledo, Micael},journal={Combinatorica},title={Chiral {Extensions} of {Regular} {Toroids}},year={2024},issn={1439-6912},month=dec,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},url={https://doi.org/10.1007/s00493-024-00132-0},urldate={2024-12-29}}
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{HMM2023_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}}
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{CM2021_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}}
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{Montero2021_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}}
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{MW2021_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}}
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{MW2021_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}}
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{Montero_2018_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}}
@phdthesis{Montero2019_ChiralExtensionsToroids_PhDThesis,author={Montero, Antonio},school={UNAM-UMSNH},title={Chiral extensions of toroids},year={2019},type={phdthesis},keywords={thesis},}
@mastersthesis{Montero2015_RealizacionesRegularesDe,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},}
@mastersthesis{Montero2013_PoliedrosRegularesEn,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
Outreach
Simetrías: Las matemáticas detrás de la belleza (Symmetry: the mathematics behind beauty)
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.
@unpublished{MP_VertexTransitiveGraphs_preprint,author={Montero, Antonio and Potočnik, Primož},note={Under review},title={Vertex-transitive graphs with small motion and transitive permutation groups with small minimal degree},month=may,year={2024},keywords={preparation},url={https://arxiv.org/abs/2405.10088}}
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 operated one, but sometimes additional symmetries are created; the same situation arises with voltage operations. We characterise the automorphisms of the operated object that are derived from the original one and use this to bound the number of flag orbits (under the action of the automorphism group) of the operated 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 operated 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.
@unpublished{HMM_SymmetriesVoltageOperations_preprint,author={Hubard, Isabel and Mochán, Elías and Montero, Antonio},note={Under review},title={Symmetries of voltage operations on polytopes, maps and maniplexes},month=dec,year={2023},keywords={preparation},relevance={relevant},url={https://arxiv.org/abs/2312.13184}}
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