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173
ON THE VASSILIEV KNOT INVARIANTS
, 1995
"... The theory of knot invariants of finite type (Vassiliev invariants) is described. These invariants turn out to be at least as powerful as the Jones polynomial and its numerous generalizations coming from various quantum groups, and it is conjectured that these invariants are precisely as powerful a ..."
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Cited by 139 (0 self)
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The theory of knot invariants of finite type (Vassiliev invariants) is described. These invariants turn out to be at least as powerful as the Jones polynomial and its numerous generalizations coming from various quantum groups, and it is conjectured that these invariants are precisely as powerful as those polynomials. As invariants of finite type are much easier to define and manipulate than the quantum group invariants, it is likely that in attempting to classify knots, invariants of finite type will play a more fundamental role than the various knot polynomials.
On the Heegaard Floer homology of branched doublecovers
 Adv. Math
"... Abstract. Let L ⊂ S 3 be a link. We study the Heegaard Floer homology of the branched doublecover Σ(L) of S 3, branched along L. When L is an alternating link, ̂HF of its branched doublecover has a particularly simple form, determined entirely by the determinant of the link. For the general case, ..."
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Cited by 58 (10 self)
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Abstract. Let L ⊂ S 3 be a link. We study the Heegaard Floer homology of the branched doublecover Σ(L) of S 3, branched along L. When L is an alternating link, ̂HF of its branched doublecover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose E 2 term is a suitable variant of Khovanov’s homology for the link L, converging to the Heegaard Floer homology of Σ(L). 1.
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.
An invariant of link cobordisms from Khovanov’s homology theory
 Algebr. Geom. Topol
"... 1.1. Khovanov’s Homology. In [K] M.Khovanov introduced a new homology theory, which assigns to a diagram D of an oriented classical link L a bigraded family of homology groups Hi,j (D) such that the graded Euler characteristic ∑ ..."
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Cited by 51 (1 self)
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1.1. Khovanov’s Homology. In [K] M.Khovanov introduced a new homology theory, which assigns to a diagram D of an oriented classical link L a bigraded family of homology groups Hi,j (D) such that the graded Euler characteristic ∑
Categorical Construction of 4D Topological Quantum Field Theories
 in Quantum Topology, L.H. Kauffman and R.A. Baadhio, eds., World Scientific
, 1993
"... In recent years, it has been discovered that invariants of three dimensional ..."
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Cited by 50 (7 self)
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In recent years, it has been discovered that invariants of three dimensional
Loops, matchings and alternatingsign matrices
 DISCR. MATH
, 2008
"... The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes on matchings is reviewed. New conjectures concerning nest distribution functions are presented as well as a bijection between certain classes of alternating sign matrices and lozenge ..."
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Cited by 32 (6 self)
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The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes on matchings is reviewed. New conjectures concerning nest distribution functions are presented as well as a bijection between certain classes of alternating sign matrices and lozenge tilings of hexagons with cut off corners.
Perturbative Topological Field Theory
 Commun. Math. Phys
"... We give a review of the application of perturbative techniques to topological quantum field theories, in particular threedimensional ChernSimonsWitten theory and its various generalizations. To this end we give an introduction to graph homology and homotopy algebras and the work of Vassiliev and K ..."
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Cited by 30 (2 self)
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We give a review of the application of perturbative techniques to topological quantum field theories, in particular threedimensional ChernSimonsWitten theory and its various generalizations. To this end we give an introduction to graph homology and homotopy algebras and the work of Vassiliev and Kontsevich on perturbative knot invariants. 1. Introduction In these lecture notes we will give a review of some recent mathematical developments in topological field theory following the work of Kontsevich, Axelrod and Singer, Vassiliev, BarNatan, Witten and others. Quite remarkable these new ideas are all related to oldfashioned perturbative techniques in field theory. Indeed, it is an interesting comment on the development of the interaction between physics and mathematics that these days pure mathematicians are calculating Feynman diagrams whereas this skill is slowly disappearing among a large fraction of theoretical physics students. Our starting point in all this will be mostly thre...
The number of primitive Vassiliev invariants up to degree 12
"... We present algorithms giving upper and lower bounds for the number of independent primitive rational Vassiliev invariants of degree m modulo those of degree m − 1. The values have been calculated for the formerly unknown degrees m = 10, 11, 12. Upper and lower bounds coincide, which reveals that all ..."
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Cited by 29 (2 self)
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We present algorithms giving upper and lower bounds for the number of independent primitive rational Vassiliev invariants of degree m modulo those of degree m − 1. The values have been calculated for the formerly unknown degrees m = 10, 11, 12. Upper and lower bounds coincide, which reveals that all Vassiliev invariants of degree ≤ 12 are orientation insensitive and are coming from representations of Lie algebras so and gl. Furthermore, a conjecture of Vogel is falsified and it is shown that the Λmodule of connected trivalent diagrams (Chinese characters) is not free. 1
Tutte polynomials and link polynomials
 Proc. Amer. Math. Soc
, 1988
"... ABSTRACT. We show how the Tutte polynomial of a plane graph can be evaluated as the "homfly " polynomial of an associated oriented link. Then we discuss some consequences for the partition function of the Potts model, the Four Color Problem and the time complexity of the computation of the homfly po ..."
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Cited by 27 (0 self)
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ABSTRACT. We show how the Tutte polynomial of a plane graph can be evaluated as the "homfly " polynomial of an associated oriented link. Then we discuss some consequences for the partition function of the Potts model, the Four Color Problem and the time complexity of the computation of the homfly polynomial. 1. Introduction. Recently
Quantum automorphism groups of small metric spaces
 Pacific J. Math
"... To any finite metric space X we associate the universal Hopf C ∗algebra H coacting on X. We prove that spaces X having at most 7 points fall into one of the following classes: (1) the coaction of H is not transitive; (2) H is the algebra of functions on the automorphism group of X; (3) X is a simpl ..."
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Cited by 25 (6 self)
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To any finite metric space X we associate the universal Hopf C ∗algebra H coacting on X. We prove that spaces X having at most 7 points fall into one of the following classes: (1) the coaction of H is not transitive; (2) H is the algebra of functions on the automorphism group of X; (3) X is a simplex and H corresponds to a TemperleyLieb algebra; (4) X is a product of simplices and H corresponds to a FussCatalan algebra.