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171
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 137 (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
A polynomial quantum algorithm for approximating the Jones polynomial
 In Proc. ACM STOC
, 2006
"... in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang provide an efficient simulation of TQFT by a quantum computer, and vic ..."
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Cited by 44 (2 self)
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in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient (namely, polynomial) quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, e 2πi/5, and moreover, that this problem is BQPcomplete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results of Freedman et al. are heavily based on TQFT, which makes the algorithm essentially inaccessible to computer scientists. We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an n strands braid with m crossings at any primitive root of unity e 2πi/k, where the running time of the algorithm is polynomial in m, n and k. Our algorithm is based, rather than on TQFT, on well known mathematical results (specifically, the path model representation of the braid group and the uniqueness of the Markov trace for the TemperleyLieb algebra). By the results of Freedman et al., our algorithm solves a BQP complete problem. Our algorithm works by encoding the local structure of the problem into the local unitary gates which are applied by the circuit. This structure is significantly different from previous quantum algorithms, which are mostly based on the Quantum Fourier
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 31 (5 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...
A polynomial invariant of oriented links
 Topology
, 1987
"... THE THEORY of classical knots and links of simple closed curves in the 3dimensional sphere has, for very many years, occupied a preeminent position in the theory of low dimensional manifolds. It has been a motivation, an inspiration and a basis for copious examples. Knots have, in theory, been cla ..."
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Cited by 30 (2 self)
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THE THEORY of classical knots and links of simple closed curves in the 3dimensional sphere has, for very many years, occupied a preeminent position in the theory of low dimensional manifolds. It has been a motivation, an inspiration and a basis for copious examples. Knots have, in theory, been classified by Haken [lo] but the classification is by means of an algorithm that is too complex to use in practice. Thus one is led to seek simple invariants for knots which will distinguish large classes of specific examples. A knot (or link) invariant is a function from the isotopy classes of knots to some algebraic structure. Perhaps the most famous invariant of a knot K is the Alexander polynomial, AK(t), a Laurent polynomial in the variable t. This was introduced by.Alexander [l] who explained how to calculate the polynomial by taking the determinant of a matrix associated with a presentation (or picture) of the knot given by a suitably chosen projection of its spatial position to a plane. The Alexander polynomial is still remarkably efficacious in distinguishing specific knots and, being readily calculable by computer, is employed by modern compilers of prime knot tables as the fundamental invariant to distinguish between examples (see Thistlethwaite [20]). Of course other invariants, notably signatures and the sophisticated CassonGordon ‘invariants’
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 27 (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