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54
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 ..."
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Cited by 55 (6 self)
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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.
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 ..."
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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.
On Triangulating ThreeDimensional Polygons
 COMPUTATIONAL GEOMETRY: THEORY AND APPLICATIONS
, 1996
"... A threedimensional polygon is triangulable if it has a nonselfintersecting triangulation which defines a simplyconnected 2manifold. We show that the problem of deciding whether a 3dimensional polygon is triangulable is NPComplete. We then establish some necessary conditions and some sufficie ..."
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Cited by 29 (3 self)
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A threedimensional polygon is triangulable if it has a nonselfintersecting triangulation which defines a simplyconnected 2manifold. We show that the problem of deciding whether a 3dimensional polygon is triangulable is NPComplete. We then establish some necessary conditions and some sufficient conditions for a polygon to be triangulable, providing special cases when the decision problem may be answered in polynomial time.
J.R.: On the LinksGould invariant of links
 J. Knot Theory Ram
"... We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra Uq[gl(21)], will be referred to as the Links–Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauff ..."
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Cited by 18 (14 self)
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We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra Uq[gl(21)], will be referred to as the Links–Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauffman polynomials (detecting chirality of some links where these invariants fail), and that it does not distinguish mutants or inverses. The method of evaluation is based on an abstract tensor state model for the invariant that is quite useful for computation as well as theoretical exploration. 1
Geometric knot spaces and polygonal isotopy
 Knots in Hellas ’98
"... Abstract. The space of nsided polygons embedded in threespace consists of a smooth manifold in which points correspond to piecewise linear or “geometric” knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n = ..."
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Cited by 11 (0 self)
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Abstract. The space of nsided polygons embedded in threespace consists of a smooth manifold in which points correspond to piecewise linear or “geometric” knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n = 6 and n = 7 is described. In both of these cases, each knot space consists of five components, but contains only three (when n = 6) or four (when n = 7) topological knot types. Therefore “geometric knot equivalence ” is strictly stronger than topological equivalence. This point is demonstrated by the hexagonal trefoils and heptagonal figureeight knots, which, unlike their topological counterparts, are not reversible. Extending these results to the cases n ≥ 8 will also be discussed.
Computing Linking Numbers of a Filtration
 In Algorithms in Bioinformatics (LNCS 2149
, 2001
"... We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of nontrivial tangling in biomolecules, modeled as alpha complexes. ..."
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Cited by 7 (5 self)
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We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of nontrivial tangling in biomolecules, modeled as alpha complexes.
Representation for knottying tasks
 in Proc. IEEE International Conference on Robotics and Automation
, 2006
"... Abstract—The learning from observation (LFO) paradigm has been widely applied in various types of robot systems. It helps reduce the work of the programmer. However, the applications of available systems are limited to manipulation of rigid objects. Manipulation of deformable objects is rarely consi ..."
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Cited by 7 (1 self)
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Abstract—The learning from observation (LFO) paradigm has been widely applied in various types of robot systems. It helps reduce the work of the programmer. However, the applications of available systems are limited to manipulation of rigid objects. Manipulation of deformable objects is rarely considered, because it is difficult to design a method for representing states of deformable objects and operations against them. Furthermore, too many operations are possible on them. In this paper, we choose knot tying as a case study for manipulating deformable objects, because the knot theory is available and the types of operations possible in knot tying are limited. We propose a knot planning from observation (KPO) paradigm, a KPO theory, and a KPO system. Index Terms—Knottying task, learning from observation (LFO), movement primitives, Reidemeister moves, state representation (Pdata). I.
THE ALMOST ALTERNATING DIAGRAMS OF THE TRIVIAL KNOT
, 2006
"... Bankwitz characterized an alternating diagram representing the trivial knot. A nonalternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknot ..."
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Cited by 5 (0 self)
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Bankwitz characterized an alternating diagram representing the trivial knot. A nonalternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknotting number one alternating knot with a property that the unknotting operation can be done on its alternating diagram.
Visualization of Seifert Surfaces
 IEEE Transactions on Visualizations and Computer Graphics
, 2006
"... Abstract—The genus of a knot or link can be defined via Seifert surfaces. A Seifert surface of a knot or link is an oriented surface whose boundary coincides with that knot or link. Schematic images of these surfaces are shown in every text book on knot theory, but from these it is hard to understan ..."
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Cited by 5 (1 self)
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Abstract—The genus of a knot or link can be defined via Seifert surfaces. A Seifert surface of a knot or link is an oriented surface whose boundary coincides with that knot or link. Schematic images of these surfaces are shown in every text book on knot theory, but from these it is hard to understand their shape and structure. In this paper, the visualization of such surfaces is discussed. A method is presented to produce different styles of surface for knots and links, starting from the socalled braid representation. Application of Seifert’s algorithm leads to depictions that show the structure of the knot and the surface, while successive relaxation via a physically based model gives shapes that are natural and resemble the familiar representations of knots. Also, we present how to generate closed oriented surfaces in which the knot is embedded, such that the knot subdivides the surface into two parts. These closed surfaces provide a direct visualization of the genus of a knot. All methods have been integrated in a freely available tool, called SeifertView, which can be used for educational and presentation purposes. 1