We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. 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.

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 1-1. 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 so-called 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.

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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. Bar-Natan 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

We study three-dimensional Chern-Simons theory with complex gauge group SL(2,C), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between the SL(2,C) partition function and the colored Jones polynomial.

We give a very short proof of the Melvin-Morton 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 four-valent tensors on the Lie algebra sl2 and the Lie superalgebra gl(1|1). This shows that the miraculous connection between the Jones and Alexander invariants follows from the similarity (supersymmetry) between sl2 and gl(1|1). 1

Abstract. We prove that the N-colored 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 A-polynomial of the torus knot can be derived from the difference equation. Also constructed is a q-hypergeometric type expression of the colored Jones polynomial forT2,2m+1. 1.

Abstract. We study relationships between the colored Jones polynomial and the A-polynomial of a knot. We establish for a large class of 2-bridge knots the AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the A-polynomial. Along the way we also calculate the Kauffman bracket skein module of all 2-bridge 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 A-polynomial of a knot, introduced in [CCGLS], describes more or less the representation

Abstract. We define a filtration on the vector space spanned by Seifert matrices of knots related to Vassiliev’s filtration on the space of knots. Further we show that the invariants of knots derived from the filtration can be expressed by coefficients of the Alexander polynomial. The theory of finite type invariants (Vassiliev invariants) for knots was first introduced by V. Vassiliev [13] and reformulated by J.S. Birman and X.S. Lin [4]. M. Kontsevich defined the universal Vassiliev invariant [9, 1] by using iterated integral. The invariant takes values in the linear combinations of chord diagrams and one can use it to construct an isomorphism from the space of all the Vassiliev invariants of degree d to the chord diagrams with d chords modulo diagrams with more chords. D. Bar-Natan [1] extended the notion of chord diagrams allowing trivalent vertices, which we call web diagrams in this paper. He showed that the space of (the linear combinations of) chord diagrams modulo the four-term relation coincides with the space of web diagrams modulo the AS, IHX and STU relations. So a main

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 uni-trivalent 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].