Results 1  10
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24
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.
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 ..."
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Cited by 18 (1 self)
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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
On the LinksGould invariant of links
 J. KNOT THEORY RAM
, 2998
"... 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.
Invariants of Knot Diagrams
 MATHEMATISCHE ANNALEN
, 2008
"... We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams. ..."
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Cited by 5 (2 self)
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We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.
Quandles knot invariants and the Nfold branched cover
, 1984
"... This paper focuses on the class of algebraic objects called involutory quandles and, in particular, on the connection between involutory quandles and topological knots and links. The paper is intended to be a selfcontained treatment of the topic and should be accessible to readers with various back ..."
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Cited by 4 (0 self)
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This paper focuses on the class of algebraic objects called involutory quandles and, in particular, on the connection between involutory quandles and topological knots and links. The paper is intended to be a selfcontained treatment of the topic and should be accessible to readers with various backgrounds.
Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle
, 2007
"... We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in R 2. 1 ..."
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Cited by 4 (1 self)
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We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in R 2. 1
The Curvature of Lattice Knots
"... A result of Milnor [1] states that the infimum of the total curvature of a tame knot K is given by 2ß¯(K), where ¯(K) is the crookedness of the knot K. It is also known that ¯(K)=b(K), where b(K) is the bridge index of K [2]. The situation appears to be quite different for knots realised as polygons ..."
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Cited by 2 (0 self)
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A result of Milnor [1] states that the infimum of the total curvature of a tame knot K is given by 2ß¯(K), where ¯(K) is the crookedness of the knot K. It is also known that ¯(K)=b(K), where b(K) is the bridge index of K [2]. The situation appears to be quite different for knots realised as polygons in the cubic lattice. We study the total curvature of lattice knots by developing algebraic techniques to estimate minimal curvature in the cubic lattice. We perform simulations to estimate the minimal curvature of lattice knots, and conclude that the situation is very different than for tame knots in R 3 . 1 Introduction How many edges are necessary and sufficient to realise a knot of type K as a polygon in the cubic lattice? This question has been addressed in references [3,4, 5,6,7]. In this paper we study a related issue: What is the minimum total curvature of a knot of type K if it is a polygon in the cubic lattice? Let Z 3 be the cubic lattice with each vertex a point in R 3 wit...
Subexponentially Computable Truncations of Jonestype Polynomials
 Graph Structure Theory, Contemporary Mathematics Vol. 147, American Mathematical Society, Providence, Rhode Island
, 1993
"... . We show that an essential part of the new (Jonestype) polynomial link invariants can be computed in subexponential time. This is in a sharp contrast to the result of Jaeger, Vertigan and Welsh that computing the whole polynomial and most of its evaluations is #P hard. 1 Introduction The discove ..."
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Cited by 1 (0 self)
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. We show that an essential part of the new (Jonestype) polynomial link invariants can be computed in subexponential time. This is in a sharp contrast to the result of Jaeger, Vertigan and Welsh that computing the whole polynomial and most of its evaluations is #P hard. 1 Introduction The discovery by V. Jones, in 1984, of a new powerful knot invariant led to a rapid growth of research in knot theory and elevated the theory of knots and links from its relative isolation. In particular, it has been noted that the "objects" similar to the Jones polynomial were studied in graph theory (the dichromatic polynomial) and statistical mechanics (e.g. the partition function for the Pott's model of antiferromagnetism) . Jones type invariants of knots are widely used: from solving old problems in topology to applications in physics, chemistry, and biology (compare [11, 37, 40]). The roots of this paper lie in a practical need for computing polynomial invariants for knots and links that have a l...