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116
Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 218 (13 self)
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For a copy with the handdrawn figures please email
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.
Threedimensional quantum gravity, ChernSimons theory, and the Apolynomial
, 2003
"... We study threedimensional ChernSimons theory with complex gauge group SL(2,C), which has many interesting connections with threedimensional quantum gravity and geometry of hyperbolic 3manifolds. We show that, in the presence of a single knotted Wilson loop in an infinitedimensional representati ..."
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Cited by 77 (10 self)
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We study threedimensional ChernSimons theory with complex gauge group SL(2,C), which has many interesting connections with threedimensional quantum gravity and geometry of hyperbolic 3manifolds. We show that, in the presence of a single knotted Wilson loop in an infinitedimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the Apolynomial of a knot. Using this approach, we find some new and rather surprising relations between the Apolynomial, the colored Jones polynomial, and other invariants of hyperbolic 3manifolds. These relations generalize the volume conjecture and the MelvinMortonRozansky conjecture, and suggest an intriguing connection between the SL(2,C) partition function and the colored Jones polynomial.
Diffeomorphisminvariant generalized measures on the space of connections modulo gauge transformations”, hepth/9305045, to appear
 in the Proceedings of the conference on quantum topology
"... The notion of a measure on the space of connections modulo gauge transformations that is invariant under diffeomorphisms of the base manifold is important in a variety of contexts in mathematical physics and topology. At the formal level, an example of such a measure is given by the ChernSimons pat ..."
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Cited by 36 (14 self)
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The notion of a measure on the space of connections modulo gauge transformations that is invariant under diffeomorphisms of the base manifold is important in a variety of contexts in mathematical physics and topology. At the formal level, an example of such a measure is given by the ChernSimons path integral. Certain measures of this sort also play the role of states in quantum gravity in Ashtekar’s formalism. These measures define link invariants, or more generally multiloop invariants; as noted by Witten, the ChernSimons path integral gives rise to the Jones polynomial, while in quantum gravity this observation is the basis of the loop representation due to Rovelli and Smolin. Here we review recent work on making these ideas mathematically rigorous, and give a rigorous construction of diffeomorphisminvariant measures on the space of connections modulo gauge transformations generalizing the recent work of Ashtekar and Lewandowski. This construction proceeds by doing lattice gauge theory on graphs analytically embedded in the base manifold. 1
The universal Rmatrix, Burau representation, and the MelvinMorton expansion of the colored Jones polynomial
 Adv. Math
, 1998
"... P. Melvin and H. Morton [9] studied the expansion of the colored Jones polynomial of a knot in powers of ˇq − 1 and color. They conjectured an upper bound on the power of color versus the power of ˇq − 1. They also conjectured that the bounding line in their expansion generated the inverse Alexander ..."
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Cited by 36 (6 self)
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P. Melvin and H. Morton [9] studied the expansion of the colored Jones polynomial of a knot in powers of ˇq − 1 and color. They conjectured an upper bound on the power of color versus the power of ˇq − 1. They also conjectured that the bounding line in their expansion generated the inverse AlexanderConway polynomial. These conjectures were proved by D. BarNatan and S. Garoufalidis [1]. We have conjectured [12] that other ‘lines ’ in the MelvinMorton expansion are generated by rational functions with integer coefficients whose denominators are powers of the AlexanderConway polynomial. Here we prove this conjecture by using the Rmatrix formula for the colored Jones polynomial and presenting Let K be a knot in S 3 endowed with canonical framing (i.e., its selflinking number is zero). We assign to this knot an αdimensional SUq(2) module Vα. Jα(K; ˇq) denotes the colored Jones polynomial of K, normalized in such a way that it is multiplicative under a
Claspers and finite type invariants of links
, 2000
"... We introduce the concept of “claspers,” which are surfaces in 3–manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called “Ck–equivalence,” which is generated by surgery operatio ..."
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Cited by 34 (3 self)
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We introduce the concept of “claspers,” which are surfaces in 3–manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called “Ck–equivalence,” which is generated by surgery operations of a certain kind called “Ck–moves”. We prove that two knots in the 3–sphere are Ck+1–equivalent if and only if they have equal values of Vassiliev–Goussarov invariants of type k with values in any abelian groups. This result gives a characterization in terms of surgery operations of the informations that can be carried by Vassiliev–Goussarov invariants. In the last section we also describe outlines of some applications of claspers to other fields in 3–dimensional topology.
A New Point of View in the Theory of Knot and Link Invariants
, 2001
"... Recent progress in string theory has led to a reformulation of quantumgroup polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to construct the new polynomials and we conjecture their general ..."
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Cited by 30 (6 self)
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Recent progress in string theory has led to a reformulation of quantumgroup polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to construct the new polynomials and we conjecture their general structure. This leads to new conjectures on the algebraic structure of the quantumgroup polynomial invariants. We also describe the geometrical meaning of the coefficients in terms of the enumerative geometry of Riemann surfaces with boundaries in a certain CalabiYau threefold.
Parametrizations of flag varieties
"... Abstract. For the flag variety G/B of a reductive algebraic group G we define and describe explicitly a certain (settheoretical) crosssection φ: G/B → G. The definition of φ depends only on a choice of reduced expression for the longest element w0 in the Weyl group W. It assigns to any gB areprese ..."
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Cited by 29 (1 self)
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Abstract. For the flag variety G/B of a reductive algebraic group G we define and describe explicitly a certain (settheoretical) crosssection φ: G/B → G. The definition of φ depends only on a choice of reduced expression for the longest element w0 in the Weyl group W. It assigns to any gB arepresentative g ∈ G together with a factorization into simple root subgroups and simple reflections. The crosssection φ is continuous along the components of Deodhar’s decomposition of G/B. We introduce a generalization of the Chamber Ansatz and give formulas for the factors of g = φ(gB). These results are then applied to parametrize explicitly the components of the totally nonnegative part of the flag variety (G/B)≥0 defined by Lusztig, giving a new proof of Lusztig’s conjectured cell decomposition of (G/B)≥0. We also give minimal sets of inequalities describing these cells. 1.
On the homology of virtual braids and the Burau representation
 arXiv:math.GT/9904089 v1 18
, 1999
"... Abstract. Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. The group of virtual braids on n strings V Bn and its Burau representation to GLnZ[t, t −1] also can be considered. The homological properties of the series of groups V Bn and its Burau representati ..."
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Cited by 28 (3 self)
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Abstract. Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. The group of virtual braids on n strings V Bn and its Burau representation to GLnZ[t, t −1] also can be considered. The homological properties of the series of groups V Bn and its Burau representation are studied. The following splitting of infinite loop spaces is proved for the plusconstruction of the classifying space of the virtual braid group on the infinite number of strings: Z × BV B + ∞ ≃ Ω ∞ S ∞ × S 1 × Y, where Y is an infinite loop space. Connections with K∗Z are discussed. 1.