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102
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 172 (15 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.
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.
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 56 (13 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.
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 40 (4 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.
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 29 (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
THE COLORED JONES POLYNOMIAL AND THE APOLYNOMIAL OF KNOTS
, 2006
"... We study relationships between the colored Jones polynomial and the Apolynomial of a knot. The AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the Apolynomial is established for a large class of twobridge knots, including all twist knots. We formulate a weaker conje ..."
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Cited by 26 (3 self)
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We study relationships between the colored Jones polynomial and the Apolynomial of a knot. The AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the Apolynomial is established for a large class of twobridge knots, including all twist knots. We formulate a weaker conjecture and prove that it holds for all twobridge knots. Along the way we also calculate the Kauffman bracket skein module of the complements of twobridge knots. Some properties of the colored Jones polynomial are established.
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 24 (3 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.
Asymptotic behaviors of the colored Jones polynomial of a torus knot
, 2004
"... We study the asymptotic behaviors of the colored Jones polynomials of torus knots. Contrary to the works by R. Kashaev, O. Tirkkonen, Y. Yokota, and the author, they do not seem to give the volumes or the Chern–Simons invariants of the threemanifolds obtained by Dehn surgeries. On the other hand i ..."
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Cited by 21 (4 self)
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We study the asymptotic behaviors of the colored Jones polynomials of torus knots. Contrary to the works by R. Kashaev, O. Tirkkonen, Y. Yokota, and the author, they do not seem to give the volumes or the Chern–Simons invariants of the threemanifolds obtained by Dehn surgeries. On the other hand it is proved that in some cases the limits give the inverse of the Alexander polynomial.
ALTERNATING SUM FORMULAE FOR THE DETERMINANT AND OTHER LINK INVARIANTS
, 2007
"... A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. We generalize this result to show that the determinant is the alternating sum of the number of quasitrees of gen ..."
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Cited by 19 (7 self)
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A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. We generalize this result to show that the determinant is the alternating sum of the number of quasitrees of genus j of the dessin of a nonalternating link. Furthermore, we obtain formulas for coefficients of the Jones polynomial by counting quantities on dessins. In particular we will show that the jth coefficient of the Jones polynomial is given by subdessins of genus less or equal to j.