. We present a formalism within which the relationship (discovered by Drinfel'd in [Dr1, Dr2]) between associators (for quasi-triangular quasi-Hopf algebras) and (a variant of) the Grothendieck-Teichmuller group becomes simple and natural, leading to a simplication of Drinfel'd's original work. In particular, we reprove that rational associators exist and can be constructed iteratively, though the proof itself still depends on the apriori knowledge that a not-necessarily-rational associator exists. Contents 1. Introduction 1 1.1. Reminders about quasi-triangular quasi-Hopf algebras 1 1.2. What we do 2 1.3. Acknowledgement 4 2. The basic denitions 4 2.1. Parenthesized braids and GT 4 2.2. Parenthesized chord diagrams and GRT 8 3. Isomorphisms and associators 11 4. The Main Theorem 15 4.1. The statement, consequences, and rst reduction 15 4.2. More on the group \ GRT 15 4.3. The second reduction 18 4.4. A cohomological interlude 19 4.5. Proof of the semi-classical hexagon equation 20...

. Using elementary counting methods, we calculate the universal perturbative invariant (also known as the LMO invariant) of a 3-manifold M , satisfying H 1 (M; Z) = Z, in terms of the Alexander polynomial of M . We show that +1 surgery on a knot in the 3-sphere induces an injective map from finite type invariants of integral homology 3-spheres to finite type invariants of knots. We also show that weight systems of degree 2m on knots, obtained by applying finite type 3m invariants of integral homology 3-spheres, lie in the algebra of Alexander-Conway weight systems, thus answering the questions raised in [Ga]. Contents 1. Introduction 1 1.1. History 1 1.2. Statement of the results 2 1.3. Acknowledgment 3 2. Preliminaries 4 2.1. Preliminaries on Chinese characters 4 2.2. The Alexander-Conway polynomial and its weight system 5 2.3. Preliminaries on the LMO invariant 7 3. Proofs 8 References 10 1. Introduction 1.1. History. In their fundamental paper, T.T.Q. Le, J. Murakami and T. Ohtsu...

This paper is part expository and part presentation of calculational results. The target space of the Kontsevich integral for knots is a space of diagrams; this space has various algebraic structures which are described here. These are utilized with Le’s theorem on the behaviour of the Kontsevich integral under cabling and with the Melvin-Morton Theorem, to obtain, in the Kontsevich integral for torus knots, both an explicit expression up to degree five and the general coefficients of the wheel diagrams.

. A formula for computing the Milnor (concordance) invariants from the Kontsevich integral is obtained. The reduced Kontsevich integral (with values in the quotient by all loop diagrams) is shown to be the universal concordance invariant of finite type. Some applications are discussed. Contents Introduction 1 1. Review of basic concepts and notation. 3 2. The vector space A(X). 6 3. The universal invariant Z(T ). 7 4. The Hopfian monoidal category quotients A h and A t . 10 5. Review of Milnor's ¯ invariants. 11 6. The reduced Kontsevich integral Z t and Milnor's invariants. 12 7. A special case. 14 8. The first non-vanishing Milnor invariants and the lattice K n (l). 15 9. Vanishing results. 17 10. The case of homotopy Milnor invariants. 18 11. Proof of Theorem 6.1. 20 12. A global formula for Milnor's invariants. 22 13. Realizing primitive diagrams. 24 14. The concordance invariance of Z t for tangles. 26 15. The universal finite type concordance invariant. 28 16. Appendix o...

Abstract. For any finite-dimensional Lie bialgebra g, we construct a bialgebra Au,v(g) over the ring C[u][[v]], which quantizes simultaneously the universal enveloping bialgebra U(g), the bialgebra dual to U(g ∗), and the symmetric bialgebra S(g). Following [Tur89], we call Au,v(g) a biquantization of S(g). We show that the bialgebra Au,v(g ∗ ) quantizing U(g ∗), U(g) ∗ , and S(g ∗ ) is essentially dual to the bialgebra obtained from Au,v(g) by exchanging u and v. Thus, Au,v(g) contains all information about the quantization of g. Our construction extends Etingof and Kazhdan’s one-variable quantization of U(g) [EK96].

Algebra, Pergamon (1970), 329--358. [Dri] V. G. Drinfeld, On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q=Q), Algebra i Analiz 2:4 (1990), 149--181. English transl.: Leningrad Math. J. 2 (1991), 829--860. [Kas] C. Kassel, Quantum groups, GTM 155, Springer-Verlag, New York 1995. [KaT] C. Kassel and V. Turaev, Chord diagram invariants of tangles and graphs, Duke Math. J. 92, no. 3 (1998), 497--552. [LM1] T. Q. T. Le and J. Murakami, Kontsevich integral for Kauffman polynomial, preprint Max-Planck-Institut Bonn, 1993. [LM2] T. Q. T. Le and J. Murakami, The universal Vassiliev-Kontsevich invariant for framed oriented links, Comp. Math. 102 (1996), 41--64. [LM3] T. Q. T. Le and J. Murakami, Parallel version of the universal Vassiliev-- Kontsevich invariant, J. Pure and Appl. Algebra 121 (1997), 271--291. [LMO] T. Q. T. Le, J. Murakami and T. Ohtsuki, On a universal perturbative invariant of 3-manifolds, Topology 37-3 (1998), 539--574. [Prz] J. H. Przytyck...

Abstract. We prove that the LMO-invariant of a 3-manifold of rank one is determined by the Alexander polynomial of the manifold, and conversely, that the Alexander polynomial is determined by the LMO-invariant. Furthermore, we show that the Alexander polynomial of a null-homologous knot in a rational homology 3-sphere can be obtained by composing the weight system of the Alexander polynomial with the ˚Arhus invariant of knots.

Abstract. Almost integral TQFT was introduced by Gilmer [G]. For each simple Lie algebra g and some prime integer we associate an almost integral TQFT which derives the projective Witten-Reshetikhin-Turaev invariant τ Pg for closed 3-manifolds. As a corollary, one can show that τ Pg is an algebraic integer for certain prime integers. The result in satisfies some Murasugi type equivalence relation if M is a homology sphere and admits a cyclic group action with fixed point set a circle. this paper can be used to prove that τ Pg M 1.

For any finite-dimensional Lie bialgebra g,we construct a bialgebra Au,v(g) over the ring C[u][[v]],which quantizes simultaneously the universal enveloping bialgebra U(g),the bialgebra dual to U(g ∗),and the symmetric bialgebra S(g). Following Turaev,we call Au,v(g) a biquantization of S(g). We show that the bialgebra Au,v(g ∗ ) quantizing U(g ∗), U(g) ∗,and S(g ∗ ) is essentially dual to the bialgebra obtained from Au,v(g) by exchanging u and v. Thus, Au,v(g) contains all information about the quantization of g. Our construction extends Etingof and Kazhdan’s one-variable quantization of U(g). Résumé. Etant donné une bigèbre de Lie g de dimension finie, nous construisons une C[u][[v]]-bigèbre Au,v(g) qui quantifie simultanément la bigèbre enveloppante U(g), la bigèbre duale de U(g ∗ ) et la bigèbre symétrique S(g). Suivant Turaev, nous appelons Au,v(g) une biquantification de S(g). Nous montrons que la bigèbre Au,v(g ∗ ) qui quantifie U(g ∗), U(g) ∗ et S(g ∗ ) est en

Abstract. We consider vector spaces Hn,ℓ and Fn,ℓ spanned by the degree-n coefficients in power series forms of the Homfly and Kauffman polynomials of links with ℓ components. Generalizing previously known formulas, we determine the dimensions of the spaces Hn,ℓ, Fn,ℓ and Hn,ℓ + Fn,ℓ for all values of n and ℓ. Furthermore, we show that for knots the algebra generated by ⊕ n Hn,1 + Fn,1 is a polynomial algebra with dim(Hn,1 + Fn,1) − 1 = n + [n/2] − 4 generators in degree n ≥ 4 and one generator in degrees 2 and 3.