Results 1  10
of
18
Triangulated manifolds with few vertices: Combinatorial manifolds
, 2005
"... Let M be a simplicial manifold with n vertices. We call M centrally symmetric if it is invariant under an involution I of its vertex set which fixes no face of M. Obviously, the number of vertices of a centrally symmetric (triangulated) manifold is even, n = 2k, and, without loss of generality, we m ..."
Abstract

Cited by 18 (1 self)
 Add to MetaCart
Let M be a simplicial manifold with n vertices. We call M centrally symmetric if it is invariant under an involution I of its vertex set which fixes no face of M. Obviously, the number of vertices of a centrally symmetric (triangulated) manifold is even, n = 2k, and, without loss of generality, we may assume that the involution is presented by the permutation I = (1 k+1)(2 k+2) · · ·(k 2k). The boundary complex ∂C ∆ k of the kdimensional crosspolytope C ∆ k is clearly centrally symmetric with respect to the standard antipodal action, and a subset F ⊆ {1, 2,...,2k} is a face of ∂C ∆ k if and only if it does not contain any minimal nonface {i, k + i} for 1 ≤ i ≤ k. Hence, every centrally symmetric manifold with 2k vertices appears as a subcomplex of the boundary complex of the kdimensional crosspolytope. Free Z2actions on spheres are at the heart of the BorsukUlam theorem, which has an abundance of applications in topology, combinatorics, functional analysis, and other areas of mathematics (see the surveys of Steinlein [50],
Isomorphism free lexicographic enumeration of triangulated surfaces and 3manifolds
, 2006
"... We present a fast enumeration algorithm for combinatorial 2 and 3manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3manifolds with 11 vertices. We further determine all equivelar maps on the nonorientable surface of genus 4 as well as a ..."
Abstract

Cited by 12 (9 self)
 Add to MetaCart
(Show Context)
We present a fast enumeration algorithm for combinatorial 2 and 3manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3manifolds with 11 vertices. We further determine all equivelar maps on the nonorientable surface of genus 4 as well as all equivelar triangulations of the orientable surface of genus 3 and the nonorientable surfaces of genus 5 and 6. 1
Combinatorial 3manifolds with 10 vertices
, 2007
"... We give a complete enumeration of all combinatorial 3manifolds with 10 vertices: There are precisely 247882 triangulated 3spheres with 10 vertices as well as 518 vertexminimal triangulations of the sphere product S 2 ×S 1 and 615 triangulations of the twisted sphere product S 2 × S 1. All the 3s ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
We give a complete enumeration of all combinatorial 3manifolds with 10 vertices: There are precisely 247882 triangulated 3spheres with 10 vertices as well as 518 vertexminimal triangulations of the sphere product S 2 ×S 1 and 615 triangulations of the twisted sphere product S 2 × S 1. All the 3spheres with up to 10 vertices are shellable, but there are 29 vertexminimal nonshellable 3balls with 9 vertices.
Generating irreducible triangulations of surfaces
, 2006
"... Abstract. Starting with the irreducible triangulations of a fixed surface and splitting vertices, all the triangulations of the surface up to a given number of vertices can be generated. The irreducible triangulations have previously been determined for the surfaces S0, S1, N1,and N2. An algorithm i ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Starting with the irreducible triangulations of a fixed surface and splitting vertices, all the triangulations of the surface up to a given number of vertices can be generated. The irreducible triangulations have previously been determined for the surfaces S0, S1, N1,and N2. An algorithm is presented for generating the irreducible triangulations of a fixed surface using triangulations of other surfaces. This algorithm has been implemented as a computer program which terminates for S1, S2, N1, N2, N3, and N4. Thus the complete sets irreducible triangulations are now also known for S2, N3, and N4, with respective cardinalities 396784, 9708, and 6297982. 1.
Surface realization with the intersection edge functional
 arXiv:math.MG/0608538
"... Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry, from the theoretical, the algorithmic, and the practical point of view. In this paper, we present a heuristic algorithm for the realization of simplicial maps, based ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry, from the theoretical, the algorithmic, and the practical point of view. In this paper, we present a heuristic algorithm for the realization of simplicial maps, based on the intersection edge functional. The heuristic was used to find geometric realizations in R 3 for all vertexminimal triangulations of the orientable surfaces of genus g = 3 and g = 4. Moreover, for the first time, examples of simplicial polyhedra in R 3 of genus 5 with 12 vertices were obtained. 1
Graph coloring manifolds
, 2005
"... We introduce a new and rich class of graph coloring manifolds via the Hom complex construction of Lovász. The class comprises examples of Stiefel manifolds, series of spheres and products of spheres, cubical surfaces, as well as examples of Seifert manifolds. Asymptotically, graph coloring manifolds ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
We introduce a new and rich class of graph coloring manifolds via the Hom complex construction of Lovász. The class comprises examples of Stiefel manifolds, series of spheres and products of spheres, cubical surfaces, as well as examples of Seifert manifolds. Asymptotically, graph coloring manifolds provide examples of highly connected, highly symmetric manifolds. 1
Necessary conditions for geometric realizability of simplicial complexes. Preprint in preparation
, 2004
"... Abstract. We associate with any simplicial complex K and any integer m a system of linear equations and inequalities. If K has a simplicial embedding in R m then the system has an integer solution. This result extends the work of I. Novik (2000). 1. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We associate with any simplicial complex K and any integer m a system of linear equations and inequalities. If K has a simplicial embedding in R m then the system has an integer solution. This result extends the work of I. Novik (2000). 1.
EXAMPLES OF PSEUDOMINIMAL TRIANGULATIONS
"... Abstract. Examples of pseudominimal triangulations on various surfaces are given. 1. ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Examples of pseudominimal triangulations on various surfaces are given. 1.
Knotted Polyhedral Tori
, 2007
"... For every knot K with stick number k there is a knotted polyhedral torus of knot type K with 3k vertices. We prove that at least 3k − 2 vertices are necessary. ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
For every knot K with stick number k there is a knotted polyhedral torus of knot type K with 3k vertices. We prove that at least 3k − 2 vertices are necessary.