Hitoshi Murakami

The colored Jones polynomials and the simplicial volume of a knot

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 [13], R.M. Kashaev defined a family of complex valued link invariants indexed by integers N ≥ 2 using the quantum dilogarithm. Later he calculated the asymptotic behavior of his invariant and observed that for the three simplest hyperbolic knots it grows as exp(Vol(K)N/2π) when N goes to the infinity, where Vol(K) is the hyperbolic volume of the complement of a knotK [14]. This amazing result and his conjecture that the same also holds for any hyperbolic knot have been almost ignored by mathematicians since his definition of the invariant is too complicated (though it uses only elementary tools). The aim of this paper is to reveal his mysterious definition and to show that his invariant is nothing but a specialization of the colored Jones polynomial. The colored Jones polynomial is defined for colored links (each component is decorated with an irreducible representation of the Lie algebra sl(2,C)). The original Jones polynomial corresponds to the case that all the colors are identical to the 2-dimensional fundamental representation. We show that Kashaev’s invariant with parameter N coincides with the colored Jones polynomial in a certain normalization with every color the N -dimensional representation, evaluated at the primitive N -th root of unity. (We have to normalize the colored Jones polynomial so that the value for the trivial knot is one, for otherwise it always vanishes). On the other hand there are other colored polynomial invariants, the generalized multivariable Alexander polynomial defined by Y. Akutsu, T. Deguchi and T. Ohtsuki [1]. They used the same Lie algebra sl(2,C) but a different hierarchy of representations. Their invariants are parameterized by c+1 parameters; an integer Date: February 1, 2008. 1991 Mathematics Subject Classification. 57M25, 57M50, 17B37, 81R50.

Quantum SO(3)-invariants dominate the SU(2)-invariant of Casson and Walker

For a compact Lie group G , E. Witten proposed topological invariants of a threemanifold (quantum G -invariants) in 1988 by using the Chern-Simons functional and the Feynman path integral [ 30 ]. See also [ 2 ]. N. Yu. Reshetikhin and V. G. Turaev gave a mathematical proof of existence of such invariants for G = SU (2) [ 28 ]. R. Kirby and P. Melvin found that the quantum SU (2)-invariant associated to q = exp(2π √ − 1/ r ) with r odd splits into the product of the quantum SO (3)-invariant and [ 15 ]. For other approaches to these invariants, see [ 3, 4, 5, 16, 22, 27 ].

Kashaev's Conjecture and the Chern-Simons Invariants of Knots and Links

R. M. Kashaev conjectured that the asymptotic behavior of the link invariant he introduced [Kashaev 951, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically that for knots 63, 89 and 820 and for the Whitehead link, the colored Jones polynomials are related to the hyperbolic volumes and the Chern-Simons invariants and propose a complexification of Kashaevs conjecture.

Invariants of three-manifolds derived from linking matrices of framed links

Introduction. In [16], R. Kirby and P. Melvin study invariants of 3manifolds τr(r>3) introduced by E. Witten [38], N. Reshetikhin and V.G. Turaev [31], and W.B.R. Lickorish [25, 26, 27] (see also [18]). In particular, Kirby and Melvin calculated τ3 and τ4 explicitly. Let M be a closed, oriented 3-manifold obtained from an (integral) framed link L. Then τ3(M) can be written as follows [16, §6].

SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial

It has been proposed that the asymptotic behavior of the colored Jones polynomial is equal to the perturbative expansion of the Chern–Simons gauge theory with complex gauge group

ASYMPTOTIC BEHAVIORS OF THE COLORED JONES POLYNOMIALS OF A TORUS KNOT

{SL(2, \mathbb{C})}

Four-genus and four-dimensional clasp number of a knot

on the hyperbolic knot complement. In this note we make the first step toward verifying this relation beyond the semi-classical approximation. This requires a careful understanding of some delicate issues, such as normalization of the colored Jones polynomial and the choice of polarization in Chern–Simons theory. Addressing these issues allows us to go beyond the volume conjecture and to verify some predictions for the behavior of the subleading terms in the asymptotic expansion of the colored Jones polynomial.

Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves

We determine all two-bridge knots with unknotting number one. In fact we prove that a two-bridge knot has unknotting number one iff there exist positive integers p, m, and n such that (, n) 1 and 2mn = p ? 1, and it is equivalent to S(p, 2n2) in Schuberts notation. It is also shown that it can be expressed as C(a, al, a2,... ,ak, +2, -ak,... ,-a2, -al) using Conways notation. Let K be a knot in a 3-sphere. An unknotting operation is an operation which changes the overcrossing and the undercrossing at a double point of a diagram of K. The unknotting number of K, denoted by u(K), is the minimum number of unknotting operations needed to deform a diagram of K into that of the trivial knot, where the minimum is taken over all diagrams of K. By a two-bridge knot S(p, q) we mean a knot which is characterized so that its double branched covering space is the lens space L(p, q), where p and q are coprime integers and p is odd and positive [3, 6, 11, 12]. (Thus we regard S(p, q) and its mirror image S(p, -q) as equivalent.) Let C(cl,C2,... Cr) be Conways notation for a two-bridge knot. If the continued fraction 1 1 C, + -2 -C is equal to p/q, then C(cl,C2,. . . ,Cr) is equivalent to S(p,q) [3, 12]. In this paper we consider two-bridge knots with unknotting number one and determine them. In fact we prove THEOREM 1. Let K be a nontrivial two-bridge knot. Then the following three conditions are equivalent. (i) u(K) = 1. (ii) There exist an odd integer p (> 1) and coprime, positive integers m and n with 2mn = p ? 1 and K is equivalent to S(p, 2n2). (iii) K can be expressed as C(a, al, a2, . . , ak, ?2,-ak, *,-a2,-al). To prove the above theorem we use the following theorem due to M. Culler, C. McA. Gordon, J. Luecke, and P. B. Shalen [4, 5] (see also Theorem A in [13]). THEOREM 2 [5]. For a knot K, let K(a/b) be a 3-manifold obtained by (a/b)Dehn surgery along K, where a and b are coprime integers. If K is not a torus knot and ri1(K(a/b)) is cyclic, then Ibl < 1. PROOF OF THEOREM 1. (i)=(ii). It is known that if a nontrivial knot K has unknotting number one then its double branched covering space is K(p/ ? 2) Received by the editors May 17, 1985. 1980 Mathematics Subject Classification. Primary 57M25.