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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 61 (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 38 (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.
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
The size of spanning disks for PL Knots.
, 1998
"... For each integer n ? 1 we construct a closed unknotted PL curve Kn in R 3 having less than 33n edges with the property that any PL triangluated disk spanning the curve contains at least 2 n triangles. 1 Introduction. We show the existence of a sequence of unknotted simple closed curves Kn in ..."
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For each integer n ? 1 we construct a closed unknotted PL curve Kn in R 3 having less than 33n edges with the property that any PL triangluated disk spanning the curve contains at least 2 n triangles. 1 Introduction. We show the existence of a sequence of unknotted simple closed curves Kn in R 3 having the following properties: ffl The curve Kn is a polygon with at most 33n edges. ffl Any PL embedding of a triangulated disk into R 3 with boundary Kn contains at least 2 n triangular faces. The existence of such disks has implications to the complexity of geometric algorithms. For example, it shows that algorithms to test knot triviality that search for embedded disks in the complement need to deal with disks containing exponentially many triangles. Thus the exponential bounds on the size of the normal disks that are analyzed in [1],[3],[4],[5], and [6] cannot be replaced with polynomial bounds. Approaches to other problems, such as the word problem for 3manifold groups,...