2012
DOI: 10.1007/s10711-012-9782-5
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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 theorems

Abstract: 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-tria… Show more

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Cited by 16 publications
(19 citation statements)
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“…In [8], a contradiction is obtained by studying the holonomy groups of such metrics, and proving the holonomy theorem which states that in a Euclidean cone metric with two cone points, of curvature ± 2π n , the holonomy group H contains the cyclic group C n of order n as a proper subgroup C n H.…”
Section: Proofs Of Nonexistence Of Almost-regular Ramification Types mentioning
confidence: 99%
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“…In [8], a contradiction is obtained by studying the holonomy groups of such metrics, and proving the holonomy theorem which states that in a Euclidean cone metric with two cone points, of curvature ± 2π n , the holonomy group H contains the cyclic group C n of order n as a proper subgroup C n H.…”
Section: Proofs Of Nonexistence Of Almost-regular Ramification Types mentioning
confidence: 99%
“…The holonomy group of a surface, M , with Euclidean cone metric is generated by π 1 (M ) together with loops around each of the cone points. It is shown in [8] that any hexangulation of M with vertices that can be two-colored has holonomy group that is a subgroup of C 3 . We can also see this by looking at the developing map into the Euclidean plane, and noticing that the tiling has a 3-symmetry.…”
Section: Proofs Of Nonexistence Of Almost-regular Ramification Types mentioning
confidence: 99%
“…of the polytope, with p max being the number of polygons with the maximum number of sides. Eberhard's theorem [23] [24]. However, there is no 13-faced simple 3-polytope with p = (0, 0, 12, 1).…”
Section: Acknowledgmentsmentioning
confidence: 99%
“…. , 3) of d. It actually turns out that this branch datum is indeed not realizable, as Zieve had conjectured, which follows from results established in [6] using geometric techniques (holonomy of Euclidean structures). The same fact was also elegantly proved by Corvaja and Zannier [3] with a more algebraic approach.…”
mentioning
confidence: 91%