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43
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 78 (9 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.
Using motion planning for knot untangling
 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH 2004; 23; 797
, 2004
"... In this paper we investigate the application of motion planning techniques to the untangling of mathematical knots. Knot untangling can be viewed as a highdimensional planning problem in reparametrizable configuration spaces. In the past, simulated annealing and other energy minimization methods ha ..."
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Cited by 31 (6 self)
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In this paper we investigate the application of motion planning techniques to the untangling of mathematical knots. Knot untangling can be viewed as a highdimensional planning problem in reparametrizable configuration spaces. In the past, simulated annealing and other energy minimization methods have been used to find knot untangling paths. We have developed a probabilistic planner that is capable of untangling knots described by over 400 variables. We have tested on known difficult benchmarks in this area and untangled them more quickly than has been achieved with minimization in the literature. In this work, the use of motion planning techniques is critical for the untangling. Our planner defines local goals and makes combined use of energy minimization and randomized treebased planning. We also show how to produce candidates with a minimal number of segments for a given knot. The planner developed in this work is novel in
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 ..."
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Cited by 17 (0 self)
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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.
The size of spanning disks for polygonal curves
 Discrete Comput. Geom
"... Abstract. For each integer n ≥ 0, there is a closed, unknotted, polygonal curve Kn in R 3 having less than 10n + 9 edges, with the property that any PiecewiseLinear triangulated disk spanning the curve contains at least 2 n−1 triangles. 1. Introduction. Let K be a closed polygonal curve in R3 consi ..."
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Cited by 12 (1 self)
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Abstract. For each integer n ≥ 0, there is a closed, unknotted, polygonal curve Kn in R 3 having less than 10n + 9 edges, with the property that any PiecewiseLinear triangulated disk spanning the curve contains at least 2 n−1 triangles. 1. Introduction. Let K be a closed polygonal curve in R3 consisting of n line segments. Assume that K is unknotted, so that it is the boundary of an embedded disk in R3. This paper considers the question: How many triangles are needed to triangulate a PiecewiseLinear (PL) spanning disk of K? The main result, Theorem 1 below,
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 10 (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
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 8 (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.
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. ..."
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Cited by 7 (6 self)
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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.
A lower bound for the number of Reidemeister moves for unknotting
 J. Knot Theory Ramif
, 2006
"... I would like to thank him for his encouragement, and letting me study anything I like when I was a student. Abstract. How many Reidemeister moves do we need for unknotting a given diagram of the trivial knot? Hass and Lagarias gave an upper bound. We give an upper bound for deforming a diagram of a ..."
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Cited by 7 (2 self)
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I would like to thank him for his encouragement, and letting me study anything I like when I was a student. Abstract. How many Reidemeister moves do we need for unknotting a given diagram of the trivial knot? Hass and Lagarias gave an upper bound. We give an upper bound for deforming a diagram of a split link to be disconnected. On the other hand, the absolute value of the writhe gives a lower bound of the number of Reidemeister I moves for unknotting. That of a complexity of knot diagram “cowrithe” works for Reidemeister II, III moves. We give an example of an infinite sequence of diagrams Dn of the trivial knot with an O(n) number of crossings such that the author expects the number of Reidemeister moves needed for unknotting it to be O(n2). However, writhe and cowrithe do not prove this. 1. An upper bound for the number of Reidemeister moves for unlinking A Reidemeister move is a local move of a link diagram as in Figure 1. Any such move does not change the link type. As Alexander and Briggs [1] and Reidemeister [7] showed that, for any pair of diagrams D1, D2 which represent the same link type, there is a finite sequence of Reidemeister moves which deforms D1 to D2. Let D be a diagram of the trivial knot. We consider sequences of Reidemeister moves which unknot D, i.e., deform D to have no crossing. Over all such sequences, we set ur(D) to be the minimal number of the moves in a sequence. Then let ur(n) denote the maximum ur(D) over all digrams of the trivial knot with n crossings. In [3], J. Hass and J. Lagarias gave an upper bound for ur(n), showing that ur(n) ≤ 2cn, where c = 1011. (See also [2].)
Virtual Knots and Links
, 2005
"... This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3dimensional topology approach that if a connected sum of two virtual knots K1 and K2 is trivial, then ..."
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Cited by 7 (0 self)
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This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3dimensional topology approach that if a connected sum of two virtual knots K1 and K2 is trivial, then so are both K1 and K2. We establish an algorithm, using HakenMatveev technique, for recognizing virtual knots. This paper may be read as both an introduction and as a research paper. For more about HakenMatveev theory and its application to classical knot theory, see [Ha, Hem, Mat, HL]. 1
The size of spanning disks for polygonal knots
, 1999
"... For each integer n ≥ 1 we construct a closed unknotted Piecewise Linear curve Kn in R 3 having less than 11n edges with the property that any Piecewise Linear triangluated disk spanning the curve contains at least 2 n−1 triangles. 1 Introduction. We show the existence of a sequence of unknotted simp ..."
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Cited by 6 (1 self)
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For each integer n ≥ 1 we construct a closed unknotted Piecewise Linear curve Kn in R 3 having less than 11n edges with the property that any Piecewise Linear triangluated disk spanning the curve contains at least 2 n−1 triangles. 1 Introduction. We show the existence of a sequence of unknotted simple closed curves Kn in R 3 having the following properties: • The curve Kn is a polygon with at most 11n edges. • Any Piecewise Linear (PL) embedding of a triangulated disk into R 3 with