2020
DOI: 10.1155/2020/5674172
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Doubly Semiequivelar Maps on Torus and Klein Bottle

Abstract: 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 … Show more

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Cited by 3 publications
(5 citation statements)
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“…Suppose, M is a map with two distinct face-sequences f 1 and f 2 . Then M is called a doubly semiequivelar map [11], in short DSEM, if (i) φ(v) has same sign (i.e., either negative, 0 or positive) for all v ∈ V (M ), and (ii) the vertices of same face-sequence also have links of same face-sequence up to a cyclic permutation. We denote the M of type [f (f 11 ,...,f 1r 1 ) 1 : f (f 21 ,...,f 2r 2 ) 2 ], where f 1i , f 2j ∈ {f 1 , f 2 }, for 1 ≤ i ≤ r 1 and 1 ≤ j ≤ r 2 , if vertices of the face-sequence f 1 have links of face-sequence (f 11 , .…”
Section: Definitions and Notationsmentioning
confidence: 99%
See 3 more Smart Citations
“…Suppose, M is a map with two distinct face-sequences f 1 and f 2 . Then M is called a doubly semiequivelar map [11], in short DSEM, if (i) φ(v) has same sign (i.e., either negative, 0 or positive) for all v ∈ V (M ), and (ii) the vertices of same face-sequence also have links of same face-sequence up to a cyclic permutation. We denote the M of type [f (f 11 ,...,f 1r 1 ) 1 : f (f 21 ,...,f 2r 2 ) 2 ], where f 1i , f 2j ∈ {f 1 , f 2 }, for 1 ≤ i ≤ r 1 and 1 ≤ j ≤ r 2 , if vertices of the face-sequence f 1 have links of face-sequence (f 11 , .…”
Section: Definitions and Notationsmentioning
confidence: 99%
“…: 3 4 .6] 1 . Their respective DSEM types are given in Table 2 of [11]. For simplicity, the types of these DSEMs are denoted by the same notations as used for the respective tilings.…”
Section: Definitions and Notationsmentioning
confidence: 99%
See 2 more Smart Citations
“…In this article, we deal with the 2-uniform tilings [3 3 .4 4 .6] 1 , [3 6 : 3 4 .6] 2 , [3 3 .4 2 : 3.4.6.4], [3 6 : 3 2 .4.12], [3.4.3.12 : 3.12 2 ], [3.4.6.4 : 4.6.12] and [3.4 2 .6 : 3.4.6.4], which induce infinitely many doubly semi-equivelar maps on torus, see [14,19]. A study of doubly semi-equivelar maps on torus and Klein bottle is also carried out in [15,16].…”
Section: Introductionmentioning
confidence: 99%