Minimal covers of equivelar toroidal maps

Drach, Kostiantyn; Mixer, Mark

doi:10.26493/1855-3974.406.3ec

Cited by 4 publications

(13 citation statements)

References 16 publications

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“…Proof. This theorem is a direct consequence of [7,Theorem 3.6]. Indeed, let τ * be the regular tessellation associated to the Archimedean tessellation τ .…”

Section: Proof Of the Main Theoremmentioning

confidence: 89%

“…There are only 11 tessellations on the plane by regular polygons so that any vertex can be mapped to every other vertex by the symmetry of the tessellation. These are the following tessellations: The equivelar Archimedean tessellations of type {3, 6}, {4, 4}, and {6, 3} and the corresponding toroidal maps were considered in [7]. In this paper we will mainly work with the non-equivelar Archimedean tessellations of the Euclidean plane; denote A to be the set of all non-equivelar tessellations.…”

Section: Preliminaresmentioning

confidence: 99%

“…In Theorems 4.2 -4.4 we will prove that each Archimedean map on the torus has a unique minimal almost regular cover on the torus, which we will construct explicitly. To accomplish these proofs, we will use some known results for equivelar toroidal maps (see [7]). Proof.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…Proof. The proof is similar to the proof of Theorem 4.2, where instead of [7,Theorem 3.6] we use [7,Theorem 3.5].…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…There is also much interest in finding minimal regular covers of different families of maps and polytopes (see for example [9,16,19]). In a previous paper [7], two of the authors constructed the minimal rotary cover of any equivelar toroidal map. Here we extend this idea to toroidal maps that are no longer equivelar, and construct minimal toroidal covers of the Archimedean toroidal maps with maximal symmetry.…”

Section: Introductionmentioning

confidence: 99%

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Archimedean toroidal maps and their minimal almost regular covers

et al. 2019

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The automorphism group of a map acts naturally on its flags (triples of incident vertices, edges, and faces). An Archimedean map on the torus is called almost regular if it has as few flag orbits as possible for its type; for example, a map of type (4.8 2 ) is called almost regular if it has exactly three flag orbits. Given a map of a certain type, we will consider other more symmetric maps that cover it. In this paper, we prove that each Archimedean toroidal map has a unique minimal almost regular cover. By using the Gaussian and Eisenstein integers, along with previous results regarding equivelar maps on the torus, we construct these minimal almost regular covers explicitly. Case 1, and is thus omitted; this completes the proof of Theorem 4.4.

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“…Proof. This theorem is a direct consequence of [7,Theorem 3.6]. Indeed, let τ * be the regular tessellation associated to the Archimedean tessellation τ .…”

Section: Proof Of the Main Theoremmentioning

confidence: 89%

Section: Preliminaresmentioning

confidence: 99%

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…Proof. The proof is similar to the proof of Theorem 4.2, where instead of [7,Theorem 3.6] we use [7,Theorem 3.5].…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Archimedean toroidal maps and their minimal almost regular covers

et al. 2019

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Minimal Covers of the Archimedean Tilings, Part II

Mixer¹,

Pellicer²,

Williams³

2013

Electron. J. Combin.

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In Part I of this paper the minimal regular covers of three Archimedean tilings were determined. However, the computations described in that work grow more complicated as the number of flag orbits of the tilings increases. In Part II, we develop a new technique in order to present the minimal regular covers of certain periodic abstract polytopes. We then use that technique to finish determining the minimal regular covers of the Archimedean tilings.

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Semi-equivelar toroidal maps and their k-semiregular covers

Kundu¹,

Maity²

2021

Preprint

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If the face-cycles at all the vertices in a map are of same type then the map is called semi-equivelar. In particular, it is called equivelar if the face-cycles contain same type of faces. A map is semiregular (or almost regular) if it has as few flag orbits as possible for its type. A map is k-regular if it is equivelar and the number of flag orbits of the map k under the automorphism group. In particular, if k = 1, its called regular. A map is k-semiregular if it contains more number of flags as compared to its type with the number of flags orbits k. Drach et al. [7] have proved that every semi-equivelar toroidal map has a finite unique minimal semiregular cover. In this article, we show the bounds of flag orbits of semi-equivelar toroidal maps, i.e., there exists k for each type such that every semi-equivelar map is ℓ-uniform for some ℓ ≤ k. We show that none of the Archimedean types on the torus is semiregular, i.e., for each type, there exists a map whose number of flag orbits is more than its type. We also prove that if a semi-equivelar map is m-semiregular then it has a finite index t-semiregular minimal cover for t ≤ m. We also show the existence and classification of n sheeted k-semiregular maps for some k of semi-equivelar toroidal maps for each n ∈ N.

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Minimal covers of equivelar toroidal maps

Cited by 4 publications

References 16 publications

Archimedean toroidal maps and their minimal almost regular covers

Archimedean toroidal maps and their minimal almost regular covers

Minimal Covers of the Archimedean Tilings, Part II

Semi-equivelar toroidal maps and their k-semiregular covers

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