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83
THE COLORED JONES POLYNOMIALS AND THE SIMPLICIAL VOLUME OF A Knot
, 1999
"... We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect nontrivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev’s quantum dilogarithm invariants for links. Theref ..."
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Cited by 101 (10 self)
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We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect nontrivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev’s quantum dilogarithm invariants for links. Therefore Kashaev’s conjecture can be restated as follows: The colored Jones polynomials determine the hyperbolic volume for a hyperbolic knot. Modifying this, we propose a stronger conjecture: The colored Jones polynomials determine the simplicial volume for any knot. If our conjecture is true, then we can prove that a knot is trivial if and only if all of its Vassiliev invariants are trivial.
A rational noncommutative invariant of boundary links
 GEOM. AND TOPOLOGY
, 2003
"... In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot. Roughly speaking, this rational presentation of the Kontsevich integral would sum formal power series into rational functions with prescribed denominators. Rozansky’s conjecture was soon pr ..."
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Cited by 42 (12 self)
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In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot. Roughly speaking, this rational presentation of the Kontsevich integral would sum formal power series into rational functions with prescribed denominators. Rozansky’s conjecture was soon proven by the first author. We begin our paper by reviewing Rozansky’s conjecture and the main ideas that lead to its proof. The natural question of extending this conjecture to links leads to the class of boundary links, and a proof of Rozansky’s conjecture in this case. A subtle issue is the fact that a “hair ” map which replaces beads by the exponential of hair is not 11. This raises the question of whether a rational invariant of boundary links exists in an appropriate space of trivalent graphs whose edges are decorated by rational functions in noncommuting variables. A main result of the paper is to construct such an invariant, using the socalled surgery view of bounadry links and after developing a formal diagrammatic Gaussian integration. Since our invariant is one of many rational forms of the Kontsevich integral, one may ask if our invariant is in some sense canonical. We prove that this is indeed the case, by axiomatically characterizing our invariant as a universal finite type invariant of boundary links with respect to the null move. Finally, we discuss relations between our rational invariant and homology surgery, and give some applications to low dimensional topology.
Remarks on the Vassiliev knot invariants coming from sl 2
 Topology
, 1997
"... this paper we give a formula for the weight system of sl 2 and describe some properties of the weight system of an arbitrary simple Lie algebra. Our formula for sl 2 is recurrent with respect to the number of chords of a diagram. The main results of this paper are Theorems 2 and 6. We will use stand ..."
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Cited by 25 (4 self)
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this paper we give a formula for the weight system of sl 2 and describe some properties of the weight system of an arbitrary simple Lie algebra. Our formula for sl 2 is recurrent with respect to the number of chords of a diagram. The main results of this paper are Theorems 2 and 6. We will use standard notations and facts about Vassiliev knot invariants mostly related to a Hopf algebra stucture on the space of chord diagrams and on the space of weight systems. For these facts and an introduction to the theory we refer to [A], [BL], [BN1], [CD], [CDL1]. In Sec.1 we recall Kontsevich's construction of weight systems with values in the universal enveloping algebras. In Sec.2 we give a recurrent formula for the weight system of sl 2 . Different formulas for this weight system where given by D. BarNatan and S. Garoufalidis in [BNG]. In Sec.3 we give new generators of the primitive part of the algebra of chord diagrams and discuss their properties. In Sec.4 we develop a language of Japanese Character Diagrams which is convenient for studying the sl 2 weight system. Proofs are in Sec.5. 1 Preliminaries
Threedimensional quantum gravity, ChernSimons theory, and the Apolynomial, preprint, arXiv: hepth/0306165
"... 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 25 (2 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.
MelvinMorton conjecture and primitive Feynman diagrams
 Internat. J. Math
, 1997
"... We give a very short proof of the MelvinMorton conjecture relating the colored Jones polynomial and the Alexander polynomial of knots. The proof is based on explicit evaluation of the corresponding weight systems on primitive elements of the Hopf algebra of chord diagrams which, in turn, follows fr ..."
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Cited by 14 (1 self)
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We give a very short proof of the MelvinMorton conjecture relating the colored Jones polynomial and the Alexander polynomial of knots. The proof is based on explicit evaluation of the corresponding weight systems on primitive elements of the Hopf algebra of chord diagrams which, in turn, follows from simple identities between fourvalent tensors on the Lie algebra sl2 and the Lie superalgebra gl(11). This shows that the miraculous connection between the Jones and Alexander invariants follows from the similarity (supersymmetry) between sl2 and gl(11). 1
A unified WittenReshetikhinTuraev invariant for integral homology spheres
, 2006
"... We construct an invariant JM of integral homology spheres M with values in a completion ̂ Z[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) WittenReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhinTurae ..."
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Cited by 14 (2 self)
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We construct an invariant JM of integral homology spheres M with values in a completion ̂ Z[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) WittenReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhinTuraev invariants of M. As a consequence, τζ(M) is an algebraic integer. Moreover, it follows that τζ(M) as a function on ζ behaves like an “analytic function ” defined on the set of roots of unity. That is, the τζ(M) for all roots of unity are determined by a “Taylor expansion ” at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, τζ(M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q = 1.
Difference equation of the colored Jones polynomial for torus knot
 Internat. J. Math
"... Abstract. We prove that the Ncolored Jones polynomial for the torus knotTs,t satisfies the second order difference equation, which reduces to the first order difference equation for a case ofT2,2m+1. We show that the Apolynomial of the torus knot can be derived from the difference equation. Also c ..."
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Cited by 13 (2 self)
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Abstract. We prove that the Ncolored Jones polynomial for the torus knotTs,t satisfies the second order difference equation, which reduces to the first order difference equation for a case ofT2,2m+1. We show that the Apolynomial of the torus knot can be derived from the difference equation. Also constructed is a qhypergeometric type expression of the colored Jones polynomial forT2,2m+1. 1.
The Colored Jones Polynomial and the APolynomial of TwoBridge Knots, preprint 2004 math.GT/0407521
"... Abstract. We study relationships between the colored Jones polynomial and the Apolynomial of a knot. We establish for a large class of 2bridge knots the AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the Apolynomial. Along the way we also calculate the Kauffman brac ..."
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Cited by 11 (3 self)
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Abstract. We study relationships between the colored Jones polynomial and the Apolynomial of a knot. We establish for a large class of 2bridge knots the AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the Apolynomial. Along the way we also calculate the Kauffman bracket skein module of all 2bridge knots. Some properties of the colored Jones polynomial of alternating knots are established. The Jones polynomial was discovered by Jones in 1994 [Jo] and has made a revolution in knot theory. Despite many efforts little is known about the relationship between the Jones polynomial and classical topology invariants like the fundamental group. The Apolynomial of a knot, introduced in [CCGLS], describes more or less the representation
Wheels, wheeling and the Kontsevich integral of the unknot
 ISRAEL J. MATH
, 1997
"... We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne [De]) for the relation between the two natural products on the space of unitrivalent diagrams. The two formulas use the related notions of “Wheels ” a ..."
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Cited by 10 (5 self)
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We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne [De]) for the relation between the two natural products on the space of unitrivalent diagrams. The two formulas use the related notions of “Wheels ” and “Wheeling”. We prove these formulas ‘on the level of Lie algebras ’ using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras. In a brief epilogue we report on recent proofs of our full conjectures, by Kontsevich [Ko2] and by DBN, DPT, and T. Q. T. Le, [BLT].