Equivelar maps on the torus

Brehm, Ulrich; Kühnel, Wolfgang

doi:10.1016/j.ejc.2008.01.010

Cited by 38 publications

(165 citation statements)

References 11 publications

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“…, 3, 4]. Brehm and Kühnel classified the equivelar maps on the 2-torus in [3]. This section considers quotients of the tesselation of tori of arbitrary rank by cubes.…”

Section: Eulerian Equivelar Quotients Of Toroidsmentioning

confidence: 99%

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Eulerian abstract polytopes

Hartley¹

2011

Aequat. Math.

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The classical theory of regular convex polytopes has inspired many combinatorial analogues. In this article, we examine two of them, the eulerian posets and the abstract regular polytopes, and see what the overlap between the concepts is. It is shown that a section regular polytope is eulerian if and only if it is spherical, or it has even rank and is locally spherical. Equivelar polytopes of rank less than 4 are eulerian, and some progress is made towards a characterisation of equivelar eulerian posets in higher rank. In particular, necessary conditions are given for an equivelar quotient of a cube or a torus to be eulerian. Mathematics Subject Classification (2000). 51M20, 20F65, 52B15.

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“…, 3, 4]. Brehm and Kühnel classified the equivelar maps on the 2-torus in [3]. This section considers quotients of the tesselation of tori of arbitrary rank by cubes.…”

Section: Eulerian Equivelar Quotients Of Toroidsmentioning

confidence: 99%

“…Together, these generate an abelian group. This abelian group has order 648 rather than 1296, since v 3 1 v 3 2 v 3 3 v 3 4 = 1. Next, let p 1 = (s 1 s 2 ) 2 , p 2 = (p 1 ) s3 and p 3 = (p 2 ) s0s1s2s1s0 .…”

Section: Schläfli Type {3 4 3 3}mentioning

confidence: 99%

Eulerian abstract polytopes

Hartley¹

2011

Aequat. Math.

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“…This observation allows us to give an easy proof of the classification of (degree-)regular tilings of the torus. This has been treated by many authors [2,16,17,5,4], although most of the results are restricted to the class of polyhedral maps.…”

Section: Euclidean Cone Metricsmentioning

confidence: 99%

“…Proof (of Theorems [1][2][3][4][5] In the situations of Theorems 1-5, the holonomy group is a subgroup of Cn with n := 6, 4, 3, 3, 2, respectively, according to Lemmas 10 and 11. On the other hand, the assumptions on the exceptional vertices imply that the equilateral metric is a euclidean cone metric with two cone points of curvature ±2π/n.…”

Section: Holonomy Groups and The Proofs Of Theorems 1-5mentioning

confidence: 99%

There is no triangulation of the torus with vertex degrees 5, 6, ... , 6, 7 and related results: geometric proofs for combinatorial theorems

et al. 2012

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There is no triangulation of the torus with vertex degrees 5, 6, . . . , 6, 7 and related results: Geometric proofs for combinatorial theoremsIvan Izmestiev · Robert B. Kusner · Günter Rote · Boris Springborn · John M. Sullivan Abstract There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We prove these results, of which the first two are known and the others seem to be new, as corollaries of a theorem on the holonomy group of a euclidean cone metric on the torus with just two cone points. We provide two proofs of this theorem: One argument is metric in nature, the other relies on the induced conformal structure and proceeds by invoking the residue theorem. Similar methods can be used to prove a theorem of Dress on infinite triangulations of the plane with exactly two irregular vertices. The non-existence results for torus decompositions provide infinite families of graphs which cannot be embedded in the torus. IntroductionIn any triangulation of the torus, the average vertex degree is 6, so vertices of degree d = 6 can be considered exceptional. It is easy to find regular triangulations with no exceptional vertices, as in Figure 1. Applying a single edge flip to such a triangulation produces a triangulation with four exceptional vertices (assuming the four vertices in question are distinct): two of degree 5 and two of degree 7, as in Figure 2(a). We call this a 5 2 7 2 -triangulation. Similarly, we can produce examples of triangulations with just two exceptional vertices, assuming these have degrees other than 5 and 7. Figure 3 shows 4,8-, 3,9-, 2,10-and 1,11-triangulations of the torus. However:The torus has no 5,7-triangulation, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7.We can also consider quadrangulations of the torus. In this case, the average vertex degree is 4, and an analogous theorem holds:Theorem 2 (Barnette, Jucovič & Trenkler [3]) The torus has no 3,5-quadrangulation, that is, no quadrangulation with exactly two exceptional vertices, of degree 3 and 5.On the other hand, 2,4-and 3 2 5 2 -quadrangulations do exist, as shown in Figure 4. Finally, one

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“…Recently, other classifications of toroidal maps have been presented, e.g. [1], [6], [11]. It is known that every equivelar toroidal map has a universal covering, in that it can be covered by a regular tessellation of the Euclidean plane.…”

Section: Introductionmentioning

confidence: 99%

Minimal covers of equivelar toroidal maps

Drach¹,

Mixer

2014

Ars Math. Contemp.

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Given any equivelar map on the torus, it is natural to consider its covering maps. The most basic of these coverings are finite toroidal maps or infinite tessellations of the Euclidean plane. In this paper, we prove that each equivelar map on the torus has a unique minimal toroidal rotary cover and also a unique minimal toroidal regular cover. That is to say, of all the toroidal rotary (or regular) maps covering a given map, there is a unique smallest. Furthermore, using the Gaussian and Eisenstein integers, we construct these covers explicitly.

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Equivelar maps on the torus

Cited by 38 publications

References 11 publications

Eulerian abstract polytopes

Eulerian abstract polytopes

There is no triangulation of the torus with vertex degrees 5, 6, ... , 6, 7 and related results: geometric proofs for combinatorial theorems

Minimal covers of equivelar toroidal maps

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