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The number of Reidemeister Moves Needed for Unknotting
, 2008
"... There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embe ..."
Abstract

Cited by 35 (11 self)
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There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embedded in the 1skeleton of the interior of a compact, orientable, triangulated PL 3manifold M. There is a positive constant c2 such that for each t ≥ 1, if M consists of t tetrahedra, and K is unknotted, then there is a sequence of at most 2 c2t elementary moves in M which transforms K to a triangle contained inside one tetrahedron of M. We obtain explicit values for c1 and c2.
Minimal Tetrahedralizations of a Class of Polyhedra
"... Given an ordinary polyhedron P in the three dimensional Euclidean space, different tetrahedralizations of P may contain different numbers of tetrahedra. Minimal tetrahedralization is a tetrahedralization with the minimum number of tetrahedra. In this paper, we present some properties of the graph of ..."
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Given an ordinary polyhedron P in the three dimensional Euclidean space, different tetrahedralizations of P may contain different numbers of tetrahedra. Minimal tetrahedralization is a tetrahedralization with the minimum number of tetrahedra. In this paper, we present some properties of the graph of polyhedra. Then we identify a class of polyhedra and show that this kind of polyhedra can be minimally tetrahedralized in O(n 2 ) time. Keywords: Computational geometry; Polyhedron; Tetrahedralization. 1 Introduction The main purpose of decomposing an object into simpler components is to simplify a problem for complex objects into a number of subproblems for simpler components. In particular, a tetrahedralization is an edge set that partitions the input domain in R 3 (i.e., three dimensional Euclidean space) into a number of tetrahedra that meet only at shared faces. The tetrahedralization has applications in the finite element method, mesh generation, CAD/CAM, computer graphics, and...