A structure theorem for pseudomanifolds

Bagchi, Bhaskar; Datta, Basudeb

doi:10.1016/s0012-365x(97)00273-2

Cited by 28 publications

(138 citation statements)

References 9 publications

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“…This implies that v ∈ lk(7) and hence lk(v) = C 7 (u, 9, 5, 4, 2, 8, 7). Then lk(7) = C 7 (8, v, u, 3, 1, 0, 6) and hence C 4 (8,5,0,7) ⊆ lk (6). So, (b, c) = (v, 4).…”

Section: This Implies That Aut(nmentioning

confidence: 98%

“…We denote this by (0, 1) (3,8) (4, u, 5, 9, 6) : (9, u, 6) ∼ = (4, 9, u). With this notation we have : (0, 1) (3,8) (4, u, 5) (6,9) : (9, u, 5) ∼ = (9, 4, u), (0, 1)(3, 8)(4, u) (5,9,6) : (9,6, u) ∼ = (5, 9, u), (0, 1)(2, 7)(3, 9) (6,8) (4, 5, u) : (9, u, 4) ∼ = (9, 5, u), (0, 1) (3, 8)(9, 6, 5) : (9,6,4) ∼ = (5, 9, 4), (0, 1)(2, 7) (3, 5, u, 4, 9) (6,8) : (9, u, 3) ∼ = (5, 9, u), (0, 1) (3, 8)(4, 9, 6, 5) : (9,6,5) ∼ = (5,4,9), considering the links of u, 4 and 5, we see that 458 or 45v is a face. Thus, lk(4) = (u, 3, 0, 5, v, z, w) or C 7 (u, 3, 0, 5, 8, z, w) for some z, w ∈ V .…”

Section: This Implies That Aut(nmentioning

confidence: 99%

“…Here M ∼ = N 3 , by the map ψ • (0, 4, 1, 8, 7) (2, u, v, 9, 5, 3). 3,9,5), i.e., lk(1) = C 7 (8,2,0,7,3,9,5). Now,02,04,17 and 19 are edges in lk(3).…”

Section: This Implies That Aut(nmentioning

confidence: 99%

“…So, if K is weakly regular and χ(K) ≥ 0 then (n, d) = (4,3), (6,4), (6,5), (12,5) (6,4), (6,5), (12,5)}, there exists a unique combinatorial 2-manifold, namely, the 4-vertex 2-sphere, the boundary of the octahedron, the 6-vertex real projective plane and the boundary of the icosahedron (cf. [5,8]). These 4 combinatorial 2-manifolds are combinatorially regular.…”

Section: Introductionmentioning

confidence: 99%

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Degree-regular triangulations of the double-torus

Datta¹,

Upadhyay²

2006

Forum Mathematicum

Self Cite

331

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“…This implies that v ∈ lk(7) and hence lk(v) = C 7 (u, 9, 5, 4, 2, 8, 7). Then lk(7) = C 7 (8, v, u, 3, 1, 0, 6) and hence C 4 (8,5,0,7) ⊆ lk (6). So, (b, c) = (v, 4).…”

Section: This Implies That Aut(nmentioning

confidence: 98%

Section: This Implies That Aut(nmentioning

confidence: 99%

“…Here M ∼ = N 3 , by the map ψ • (0, 4, 1, 8, 7) (2, u, v, 9, 5, 3). 3,9,5), i.e., lk(1) = C 7 (8,2,0,7,3,9,5). Now,02,04,17 and 19 are edges in lk(3).…”

Section: This Implies That Aut(nmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Degree-regular triangulations of the double-torus

Datta¹,

Upadhyay²

2006

Forum Mathematicum

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331

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“…, u 12 u 23 u 34 u 45 u 51 : 1 ≤ i ≤ 5}, RP 2 6 = {uu i u i+1 , u i u i+1 u i+3 : 1 ≤ i ≤ 5}, R = {u 1,2 u 2,3 u 3,4 u 4,5 u 5,1 , u i,i+1 u i,i+1,i+3 u i+2,i+3,i u i+5,i,i+2 u i+5,i : 1 ≤ i ≤ 5}, M 1 = {u 1+ p u 4+ p u 7+ p , u i+3 p u j+3 p u k+3 p : (i, j, k) ∈ {(1, 2, 5), (1,3,5), (1,3,4), (1,8,9), (1,6,8), (1,2,6), (2, 3, 6)},…”

Section: Introductionmentioning

confidence: 99%

Equivelar Polyhedra with Few Vertices

Datta

Nilakantan

2001

Discrete Comput Geom

727

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We know that the polyhedra corresponding to the Platonic solids are equivelar. In this article we have classified completely all the simplicial equivelar polyhedra on ≤ 11 vertices. There are exactly 27 such polyhedra. For each n ≥ −4, we have classified all the ( p, q) such that there exists an equivelar polyhedron of type { p, q} and of Euler characteristic n. We have also constructed five types of equivelar polyhedra of Euler characteristic −2m, for each m ≥ 2.

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Degree-regular triangulations of torus and Klein bottle

2005

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A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices.In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists an n-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinct n-vertex weakly regular triangulations of the torus for each n ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for each m ≥ 2. For 12 ≤ n ≤ 15, we have classified all the n-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.Proposition 1 . There are exactly 77 weakly regular combinatorial 2-manifolds on at most 15 vertices; 42 of these are orientable and 35 are non-orientable. Among these 77 combinatorial 2-manifolds, 20 are of Euler characteristic 0. These 20 are T

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A structure theorem for pseudomanifolds

Cited by 28 publications

References 9 publications

Degree-regular triangulations of the double-torus

Degree-regular triangulations of the double-torus

Equivelar Polyhedra with Few Vertices

Degree-regular triangulations of torus and Klein bottle

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