Results 1  10
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10
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 ..."
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Cited by 7 (5 self)
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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
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 ..."
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Cited by 5 (3 self)
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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
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 ..."
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Cited by 5 (0 self)
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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.
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 ..."
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Cited by 4 (0 self)
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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
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 ..."
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Cited by 4 (2 self)
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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.
Degreeregular triangulations of the doubletorus
, 2005
"... A connected combinatorial 2manifold is called degreeregular if each of its vertices have the same degree. A connected combinatorial 2manifold is called weakly regular if it has a vertextransitive automorphism group. Clearly, a weakly regular combinatorial 2manifold is degreeregular and a degre ..."
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Cited by 3 (1 self)
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A connected combinatorial 2manifold is called degreeregular if each of its vertices have the same degree. A connected combinatorial 2manifold is called weakly regular if it has a vertextransitive automorphism group. Clearly, a weakly regular combinatorial 2manifold is degreeregular and a degreeregular combinatorial 2manifold of Euler characteristic −2 must contain 12 vertices. In 1982, McMullen et al. constructed a 12vertex geometrically realized triangulation of the doubletorus in R 3. As an abstract simplicial complex, this triangulation is a weakly regular combinatorial 2manifold. In 1999, Lutz showed that there are exactly three weakly regular orientable combinatorial 2manifolds of Euler characteristic −2. In this article, we classify all the orientable degreeregular combinatorial 2manifolds of Euler characteristic −2. There are exactly six such combinatorial 2manifolds. This classifies all the orientable equivelar polyhedral maps of Euler characteristic −2. AMS classification: 57Q15, 57M20, 57N05.
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. ..."
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Cited by 3 (0 self)
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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.
IsomorphismFree Lexicographic Enumeration of Triangulated Surfaces and 3Manifolds
, 2007
"... 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 polyhedral maps on the nonorientable surface of genus 4 a ..."
Abstract
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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 polyhedral 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
Surface Realization with the Intersection Segment Functional
, 2009
"... 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
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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 segment 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
unknown title
, 707
"... 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. 1 ..."
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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. 1