arXiv subjects
Publications and source records attributed to Jun Murakami.
The present paper gives two concrete formulas for the volume of an arbitrary spherical tetrahedron, which is in a 3-dimensional spherical space of constant curvature +1. One formula is given in terms of dihedral angles, and another one is given in terms of edge lengths.
We construct knot invariants from the radical part of projective modules of restricted quantum groups. We also show a relation between these invariants and the colored Alexander invariants.
We give a closed formula for volumes of generic hyperbolic tetrahedra in terms of edge lengths. The cue of our formula is by the volume conjecture for the Turaev-Viro invariant of closed 3-manifolds, which is defined from the quantum 6j-symbols. This formula contains the dilogarithm functions, and we specify the adequate branch to get the actual value of the volumes.
R.M. Kashaev conjectured that the asymptotic behavior of his link invariant, 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 $6_3$, $8_9$ and $8_{20}$ 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 Kashaev's conjecture.
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.
We construct an invariant of 3-manifolds using a modification of the Kontsevich integral and Kirby's calculus. This invariant, as expected in perturbative Chern-Simon theory, takes values in the algebra of oriented 3-valent graphs. This algebra is a Hopf algebra, graded by half the number of vertices in 3-valent graphs. The degree 1 term of the invariant coincides with Casson-Walker-Lescop invariant. The degree $n$ term is constructed out of the universal Vassiliev invariant of links of degree less than or equal to $(l+1)n$ where $l$ is the number of link components.
We give a generalization of the Reshetikhin-Turaev functor for tangles to get a combinatorial formula for the universal Vassiliev-Kontsevich invariant of framed oriented links which is coincident with the Kontsevich integral. The universal Vassiliev-Kontsevich invariant is constructed using the Drinfeld associator. We prove the uniqueness of the Drinfeld associator. As a corollary one gets the rationality of the Kontsevich integral. Many properties of the universal Vassiliev-Kontsevich invariant are established. Connections to quantum group invariants and to multiple zeta values are discussed.