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14
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.
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 44 (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.
A New Algorithm For Recognizing The Unknot
 Geometry and Topology
, 2000
"... The topological underpinnings are presented for a new algorithm which answers the question: \Is a given knot the unknot?" The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that ..."
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Cited by 16 (2 self)
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The topological underpinnings are presented for a new algorithm which answers the question: \Is a given knot the unknot?" The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and only if it is the boundary of a disc with a combinatorial foliation. The main problems which are solved in this paper are: how to systematically enumerate combinatorial braid foliations of a disc; how to verify whether a combinatorial foliation can be realized by an embedded disc; how to nd a word in the the braid group whose conjugacy class represents the boundary of the embedded disc; how to check whether the given knot is isotopic to one of the enumerated examples; and nally, how to know when we can stop checking and be sure that our example is not the unknot. AMS Classication numbers Primary: 57M25, 57M50, 68Q15 Secondary: 57M15, 68U05 Keywords: knot, unknot, braid, foliation, algorithm Published 4 January 1999, Geometry and Topology, Volume 2 (1998) 175220 1 Contents 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,
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
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. ..."
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Cited by 6 (3 self)
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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.
Towards an implementation of the BH algorithm for recognizing the unknot
 In KNOTS2000
, 2001
"... In the manuscript [2] the rst author and Michael Hirsch presented a thennew algorithm for recognizing the unknot. The rst part of the algorithm required the systematic enumeration of all discs which support a `braid foliation' and are embeddable in 3space. The boundaries of these `foliated ..."
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Cited by 5 (3 self)
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In the manuscript [2] the rst author and Michael Hirsch presented a thennew algorithm for recognizing the unknot. The rst part of the algorithm required the systematic enumeration of all discs which support a `braid foliation' and are embeddable in 3space. The boundaries of these `foliated embeddable discs' (FED's) are the collection of all closed braid representatives of the unknot, up to conjugacy, and the second part of the algorithm produces a word in the generators of the braid group which represents the boundary of the previously listed FED's. The third part tests whether a given closed braid is conjugate to the boundary of a FED on the list. In this paper we describe implementations of the rst and second parts of the algorithm. We also give some of the data which we obtained. The data suggests that FED's have unexplored and interesting structure. Open questions are interspersed throughout the manuscript. The third part of the algorithm was studied in [3] and [4], and implemented by S.J. Lee [20]. At this writing his algorithm is polynomial for n 4 and exponential for n 5. 1
SIMPLICIAL STRUCTURES OF KNOT COMPLEMENTS
, 2003
"... It was shown in [5] that there exists an explicit bound for the number of Pachner moves needed to connect any two triangulation of any Haken 3manifold which contains no fibred submanifolds as strongly simple pieces of its JSJdecomposition. In this paper we prove a generalisation of that result to ..."
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Cited by 3 (0 self)
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It was shown in [5] that there exists an explicit bound for the number of Pachner moves needed to connect any two triangulation of any Haken 3manifold which contains no fibred submanifolds as strongly simple pieces of its JSJdecomposition. In this paper we prove a generalisation of that result to all knot complements. The explicit formula for the bound is in terms of the numbers of tetrahedra in the two triangulations. This gives a conceptually trivial algorithm for recognising any knot complement among all 3manifolds.
TRIANGULATIONS OF SEIFERT FIBRED MANIFOLDS
, 2003
"... It is not completely unreasonable to expect that a computable function bounding the number of Pachner moves needed to change any triangulation of a given 3manifold into any other triangulation of the same 3manifold exists. In this paper we describe a procedure yielding an explicit formula for suc ..."
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Cited by 2 (0 self)
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It is not completely unreasonable to expect that a computable function bounding the number of Pachner moves needed to change any triangulation of a given 3manifold into any other triangulation of the same 3manifold exists. In this paper we describe a procedure yielding an explicit formula for such a function if the 3manifold in question is a Seifert fibred space.
A new algorithm for recognizing the unknot
, 2008
"... The topological underpinnings are presented for a new algorithm which answers the question: “Is a given knot the unknot?” The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot ..."
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Cited by 1 (0 self)
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The topological underpinnings are presented for a new algorithm which answers the question: “Is a given knot the unknot?” The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and only if it is the boundary of a disc with a combinatorial foliation. The main problems which are solved in this paper are: how to systematically enumerate combinatorial braid foliations of a disc; how to verify whether a combinatorial foliation can be realized by an embedded disc; how to find a word in the the braid group whose conjugacy class represents the boundary of the embedded disc; how to check whether the given knot is isotopic to one of the enumerated examples; and finally, how to know when we can stop checking and be sure that our example is not the unknot.