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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 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.
Quaternionic invariants of virtual knots and links
"... AMS classification 57M27 In this paper we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2×2 matrices with entries in a possibly noncommutative ring, for example the quaternions. These polynomials are sufficiently powerful ..."
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Cited by 16 (8 self)
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AMS classification 57M27 In this paper we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2×2 matrices with entries in a possibly noncommutative ring, for example the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from any classical knot, including the unknot.
Numerical Simulation of Gel Electrophoresis of DNA Knots in Weak and Strong Electric Fields
, 2006
"... This unedited manuscript has been accepted for publication in Biophysical ..."
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This unedited manuscript has been accepted for publication in Biophysical
GENERALIZED QUATERNIONS and INVARIANTS of VIRTUAL KNOTS and LINKS
, 2006
"... In this paper we show how generalized quaternions including 2×2 matrices can be used to find solutions of the equation [B,(A − 1)(A,B)] = 0. These solutions can then be used to find polynomial invariants of virtual knots and links. ..."
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In this paper we show how generalized quaternions including 2×2 matrices can be used to find solutions of the equation [B,(A − 1)(A,B)] = 0. These solutions can then be used to find polynomial invariants of virtual knots and links.
Classical Roots of Knot Theory
"... AbstractVandermonde wrote in 1771: “Whatever the twists and turns of a system of threads in space, one can always obtain an expression for the calculation of its dimensions, but this expression will be of little use in practice. The craftsman who fashions a braid, a net, or some knots will be conce ..."
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AbstractVandermonde wrote in 1771: “Whatever the twists and turns of a system of threads in space, one can always obtain an expression for the calculation of its dimensions, but this expression will be of little use in practice. The craftsman who fashions a braid, a net, or some knots will be concerned, not with questions of measurement, but with those of position: what he sees there is the manner in which the threads are interlaced. ” We sketch in this essay the history of knot theory stressing the
2.1 The Alexander Invariant....................... 2
, 2012
"... In this paper we prove the validity of a new algorithm for computing the Alexander invariant, which was originally conjectured by BarNatan ..."
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In this paper we prove the validity of a new algorithm for computing the Alexander invariant, which was originally conjectured by BarNatan