Results 1  10
of
21
The computational Complexity of Knot and Link Problems
 J. ACM
, 1999
"... We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting pr ..."
Abstract

Cited by 55 (6 self)
 Add to MetaCart
We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without selfintersection 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 PSPACE. We also give exponential worstcase 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.
Computational Topology: Ambient Isotopic Approximation of 2Manifolds
 Theoretical Computer Science
, 2001
"... A fundamental issue in theoretical computer science is that of establishing unambiguous formal criteria for algorithmic output. This paper does so within the domain of computeraided geometric modeling. For practical geometric modeling algorithms, it is often desirable to create piecewise linear appr ..."
Abstract

Cited by 28 (14 self)
 Add to MetaCart
A fundamental issue in theoretical computer science is that of establishing unambiguous formal criteria for algorithmic output. This paper does so within the domain of computeraided geometric modeling. For practical geometric modeling algorithms, it is often desirable to create piecewise linear approximations to compact manifolds embedded in and it is usually desirable for these two representations to be "topologically equivalent". Though this has traditionally been taken to mean that the two representations are homeomorphic, such a notion of equivalence suffers from a variety of technical and philosophical difficulties; we adopt the stronger notion of ambient isotopy. It is shown here, that for any , compact, 2manifold without boundary, which is embedded in , there exists a piecewise linear ambient isotopic approximation. Furthermore, this isotopy has compact support, with specific bounds upon the size of this compact neighborhood. These bounds may be useful for practical application in computer graphics and engineering design simulations. The proof given relies upon properties of the medial axis, which is explained in this paper. Email address: amenta@cs.utexas.edu. This author acknowledges, with appreciation, funding from the National Science Foundation under grant number CCR9731977, and a research fellowship from the Alfred P. Sloan Foundation. The views expressed herein are those of the author, not of these sponsors.
Interactive Topological Drawing
, 1998
"... The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues ..."
Abstract

Cited by 18 (1 self)
 Add to MetaCart
The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in threedimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or selfintersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a
The size of triangulations supporting a given link
, 2000
"... Abstract. Let T be a triangulation of S 3 containing a link L in its 1skeleton. We give an explicit lower bound for the number of tetrahedra of T in terms of the bridge number of L. Our proof is based on the theory of almost normal surfaces. 1. ..."
Abstract

Cited by 8 (6 self)
 Add to MetaCart
Abstract. Let T be a triangulation of S 3 containing a link L in its 1skeleton. We give an explicit lower bound for the number of tetrahedra of T in terms of the bridge number of L. Our proof is based on the theory of almost normal surfaces. 1.
The computational complexity of knot genus and spanning area
 electronic), arXiv: math.GT/0205057. MR MR2219001
"... Abstract. We show that the problem of deciding whether a polygonal knot in a closed threedimensional manifold bounds a surface of genus at most g is NPcomplete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant C is NPha ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Abstract. We show that the problem of deciding whether a polygonal knot in a closed threedimensional manifold bounds a surface of genus at most g is NPcomplete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant C is NPhard. 1.
Algorithms for recognizing knots and 3manifolds
 Chaos, Solitons and Fractals
, 1998
"... Algorithms are of interest to geometric topologists for two reasons. First, they have bearing on the decidability of a problem. Certain topological questions, such as finding a classification of four dimensional manifolds, admit no solution. ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Algorithms are of interest to geometric topologists for two reasons. First, they have bearing on the decidability of a problem. Certain topological questions, such as finding a classification of four dimensional manifolds, admit no solution.
Emerging Challenges in Computational Topology
 Results of the NFS Workshop on Computational Topology
, 1999
"... Here we present the results of the NSFfunded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology. Author affiliations: Marshall Bern, Xerox Palo Alto Research Ctr., bern@parc. ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Here we present the results of the NSFfunded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology. Author affiliations: Marshall Bern, Xerox Palo Alto Research Ctr., bern@parc.xerox.com. David Eppstein, Univ. of California, Irvine, Dept. of Information & Computer Science, eppstein@ics.uci.edu. Pankaj K. Agarwal, Duke Univ., Dept. of Computer Science, pankaj@cs.duke.edu. Nina Amenta, Univ. of Texas, Austin, Dept. of Computer Sciences, amenta@cs.utexas.edu. Paul Chew, Cornell Univ., Dept. of Computer Science, chew@cs.cornell.edu. Tamal Dey, Ohio State Univ., Dept. of Computer and Information Science, tamaldey@cis.ohiostate.edu. David P. Dobkin, Princeton Univ., Dept. of Computer Science, dpd@cs.princeton.edu. Herbert Edelsbrunner, Duke Univ., Dept. of Computer Science, edels@cs.duke.edu. Cindy Grimm, Brown Univ., Dept. of Computer Science, cmg@cs.brown.edu. Leonid...
Euclidicity Criteria For ThreeDimensional Branched Triangulations
, 1994
"... The subject of this paper arises from the study of equivalence classes of equivariant periodic tilings of euclidean threespace. An equivariant tiling is just an ordinary tiling, i.e. a subdivision of a space into a set of topological balls, together with a group of symmetries which are compatible w ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
The subject of this paper arises from the study of equivalence classes of equivariant periodic tilings of euclidean threespace. An equivariant tiling is just an ordinary tiling, i.e. a subdivision of a space into a set of topological balls, together with a group of symmetries which are compatible with this subdivision. Such tilings are called equivalent or of the same type if there exists a homeomorphism between the underlying spaces which respects the subdivisions on both sides and also maps one symmetry group bijectively onto the other. The special case we are interested in here are tilings of euclidean threespace with associated symmetry groups which are discrete groups of euclidean motions with compact orbit spaces, i.e. which are socalled crystallographic space groups...
Structures of small closed nonorientable 3manifold triangulations
"... A census is presented of all closed nonorientable 3manifold triangulations formed from at most seven tetrahedra satisfying the additional constraints of minimality and P 2irreducibility. The eight different 3manifolds represented by these 41 different triangulations are identified and the combin ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
A census is presented of all closed nonorientable 3manifold triangulations formed from at most seven tetrahedra satisfying the additional constraints of minimality and P 2irreducibility. The eight different 3manifolds represented by these 41 different triangulations are identified and the combinatorial structures of the triangulations are described in detail. Furthermore, infinite families of triangulations are constructed that highlight the common features of the census triangulations and allow similar triangulations of additional 3manifolds to be formed. Algorithms and techniques used in constructing the census are included. 1