498 research outputs found
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
Strategies for optimization of hexahedral meshes and their comparative study
In this work, we study several strategies based on different objective functions for optimization of hexahedral meshes. We consider two approaches to construct objective functions. The first one is based on the decomposition of a hexahedron into tetrahedra. The second one is derived from the Jacobian matrix of the trilinear mapping between the reference and physical hexahedral element. A detailed description of all proposed strategies is given in the present work. Some computational experiments have been developed to test and compare the untangling capabilities of the considered objective functions. In the experiments, a sample of highly distorted hexahedral elements is optimized with the proposed objective functions, and the rate of success of each function is obtained. The results of these experiments are presented and analyzed.SecretarĂa de Estado de Universidades e InvestigaciĂłn del Ministerio de EconomĂa y Competitividad del Gobierno de España; Programa de FPU del Ministerio de EducaciĂłn, Cultura y Deporte; Programa de
FPI propio de la ULPGC; Fondos FEDE
The number of Reidemeister Moves Needed for Unknotting
There is a positive constant such that for any diagram representing
the unknot, there is a sequence of at most Reidemeister moves that
will convert it to a trivial knot diagram, is the number of crossings in
. A similar result holds for elementary moves on a polygonal knot
embedded in the 1-skeleton of the interior of a compact, orientable,
triangulated 3-manifold . There is a positive constant such that
for each , if consists of tetrahedra, and is unknotted,
then there is a sequence of at most elementary moves in which
transforms to a triangle contained inside one tetrahedron of . We obtain
explicit values for and .Comment: 48 pages, 14 figure
Triangulations
The earliest work in topology was often based on explicit combinatorial models – usually triangulations – for the spaces being studied. Although algebraic methods in topology gradually replaced combinatorial ones in the mid-1900s, the emergence of computers later revitalized the study of triangulations. By now there are several distinct mathematical communities actively doing work on different aspects of triangulations. The goal of this workshop was to bring the researchers from these various communities together to stimulate interaction and to benefit from the exchange of ideas and methods
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