Archimedean solids of genus two

Karabáš, Ján; Nedela, Roman

doi:10.1016/j.endm.2007.01.047

Cited by 9 publications

(111 citation statements)

References 5 publications

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“…On the other hand when lk(a 4 ) � C 6 (a 2 , 5, 8, a 3 , 7, 6), we get (n 13 , n 14 , n 15 ) ∈{(2, 1, 5), (1,5,8), (3,8,5), (5, 1, 2), (5, 8, 3), (8, 5, 1)}. If (n 13 , n 14 , n 15 ) � (2, 1, 5) and (1,5,8), then lk(7) is a cycle of length 5 and 6 respectively, which is not possible. If (n 13 , n 14 , n 15 ) � (3, 8, 5) and (5, 1, 2), then we see easily that lk(7) and lk(8) cannot be completed, respectively.…”

Section: Computation and Classification For Typementioning

confidence: 98%

“…When (x 1 , x 2 ) � (a 6 , a 5 ), then successively we get lk(a 2 ) � C 6 (a 1 , a 3 , a 6 , a 5 , 3, 2), lk(a 5 ) � C 6 (a 1 , a 4 , a 6 , a 2 , 3, 1), lk(a 4 ) � C 6 (a 3 , a 1 , a 5 , a 6 , 5, 4), lk(a 3 ) � C 6 (a 4 , a 1 , a 2 , a 6 , 6, 4), lk(a 6 ) � C 6 (a 3 , a 2 , a 5 , a 4 , 5, 6). Considering lk(1), it is easy to see that (n 2 , n 3 , n 4 ) ∈{(4, 5, 6), (4,6,5), (5,4,6), (5,6,4), (6,4,5), (6,5,4) a 1 , a 2 ) (a 4 , a 6 ). So, we search for (n 2 , n 3 , n 4 ) ∈ {(4, 5, 6), (4,6,5), (5,4,6)}.…”

Section: Computation and Classification For Typementioning

confidence: 99%

“…Equivelar and semiequivelar maps are generalizations of the maps on the surfaces of well-known Platonic solids and Archimedean solids to the closed surfaces other than the 2sphere, respectively. A substantial literature is available for such maps (see [1][2][3][4][5][6][7][8]).…”

Section: Introductionmentioning

confidence: 99%

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Doubly Semiequivelar Maps on Torus and Klein Bottle

Tiwari

Tripathi²,

Singh

et al. 2020

Journal of Mathematics

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A tiling of the Euclidean plane, by regular polygons, is called 2-uniform tiling if it has two orbits of vertices under the action of its symmetry group. There are 20 distinct 2-uniform tilings of the plane. Plane being the universal cover of torus and Klein bottle, it is natural to ask about the exploration of maps on these two surfaces corresponding to the 2-uniform tilings. We call such maps as doubly semiequivelar maps. In the present study, we compute and classify (up to isomorphism) doubly semiequivelar maps on torus and Klein bottle. This classification of semiequivelar maps is useful in classifying a category of symmetrical maps which have two orbits of vertices, named as 2-uniform maps.

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Section: Computation and Classification For Typementioning

confidence: 98%

Section: Computation and Classification For Typementioning

confidence: 99%

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Doubly Semiequivelar Maps on Torus and Klein Bottle

Tiwari

Tripathi²,

Singh

et al. 2020

Journal of Mathematics

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“…This approach involves actions of groups on surfaces. They also present in [13] an atlas of the so-called Archimedean solids on an orientable surface with Euler characteristic −2.…”

Section: Introduccionmentioning

confidence: 99%

Vertex-transitive maps with Schläfli type{3,7}

Pellicer

2014

Discrete Mathematics

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Among all equivelar vertex-transitive maps on a given closed surface S, the automorphism groups of maps with Schläfli types {3, 7} and {7, 3} allow the highest possible order. We describe a procedure to transform all such maps into 1-or 2-orbit maps, whose symmetry type has been previously studied. In so doing we provide a procedure to determine all vertex-transitive maps with Schläfli type {3, 7} which are neither regular or chiral. We determine all such maps on surfaces with Euler characteristic −1 ≥ χ ≥ −40.

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“…The main result of our paper written in the enumerative form follows. Complete lists of Archimedean maps of genus 2, 3, 4 can be found at the website [18]. …”

Section: Two Links {X Xl} and {Y Yl} Are Parallel If I(x) = I(y) mentioning

confidence: 99%

Archimedean maps of higher genera

2012

Self Cite

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Abstract. The paper focuses on the classification of vertex-transitive polyhedral maps of genus from 2 to 4. These maps naturally generalise the spherical maps associated with the classical Archimedean solids. Our analysis is based on the fact that each Archimedean map on an orientable surface projects onto a one-or a two-vertex quotient map. For a given genus g ≥ 2 the number of quotients to consider is bounded by a function of g. All Archimedean maps of genus g can be reconstructed from these quotients as regular covers with covering transformation group isomorphic to a group G from a set of g-admissible groups. Since the lists of groups acting on surfaces of genus 2, 3 and 4 are known, the problem can be solved by a computer-aided case-to-case analysis.

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Archimedean solids of genus two

Cited by 9 publications

References 5 publications

Doubly Semiequivelar Maps on Torus and Klein Bottle

Doubly Semiequivelar Maps on Torus and Klein Bottle

Vertex-transitive maps with Schläfli type{3,7}

Archimedean maps of higher genera

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