Kashaev intoruduced a knot invariant which is conjedtured to relate to the hyperbolic volume of the complement of a hyperbolic knot. In this paper show that Kashaev's invariant is equal to a specialization of the coloered Jones polynomial. This means that the colored Jones polynomial may relate to the hyperolic volume of the complement of a hyperbolic knot. We extend this conjecture to general knots by using the Gromov's simplicial volume.
Knot Thoery -dedicated to Professor Kunio Murasugi for his 70th birthday- Toronto July 13th - 17th, 1999. Editors: M. Sakuma et al., Osaka University, March 2000, pp. 258-267.
We show that there is no finite-type invariant of degree
less than 11 which distinguishes mutant knots.
A three-manifold invariant via the Kontsevich integral (with Thang T. Q. Le, Hitoshi Murakami and Tomotada Ohtsuki)
Osaka J. Math. 36 (2) (1999), 365-396.
In this paper, we show a relation of the Kontsevich invariant concerning to the second Kirby move of links, which is important to construct the universal perturvative invariant.
We also study the degree 1 part of the universal perturvative invariant concretely and showed that this part coinsides with the Lescop's generalization of the Casson-Walker invariant.
On a universal perturbative invariant of 3-manifolds (with Thang T. Q. Le and Tomotada Ohtsuki)
Topology 37 (1998), 539-574.
In this paper, we constructed the universal perturbative invariant of 3-manifolds from the Kontsevich invariant of links. T. T. Q. Le showed that the universal perturbative invariant is universal for the finite type invariant of rational homology spheres introduced by T. Ohtsuki and S. Garoufaridis.
T. Ohtsuki showed that the perturbative SO(3) invariant of rational homology spheres is recovered from the universal perturbative invariant.
Topological quantum field theory for the universal quantum invariant (with T. Ohtsuki)
The free-fermion model in presence of field related to the quantum group $U_q(sl_2)$ of affine type and the multi-variable Alexander polynomial of links
Infinite analysis, Advanced Series in Mathmatical Physics, 16 B (1991), 765-772.
The representations of the q-analogue of Brauer's centralizer algebras and the Kauffman polynomial of links
Publ. Res. Inst. Math. Sci. 26 (1990), 935-945.
Solvable lattice models and algebras of face operators
Integrable systems in quantum field theory and statistical mechanics, Advanced Studies in Pure Mathematics, 19 (1989), 399-415, Academic Press.
The parallel version of polynomial invariants of links (Dr. Thesis)
Osaka J. Math. 26 (1989), 1-55.
Define parallel versions of the Jones polynomial and its generalizations and study their properties by using representations of the braid groups to some algebras. As a result, it turns out that the 3-parallel version of the HOMFLY polynomial can distinguish two mutant knots (if they are "general").
Cyclotomic invariants for links (with Tsuyoshi Kobayashi and Hitoshi Murakami)
Proc. Japan Academy, Ser. A. 64 (1988), no,7, 235-238.
The Kauffman polynomial of links and representation theory
Osaka J. Math. 24 (1987), 745-758.
As in the case of Jones polynomial, The Kauffman polynomial of knots and links is expressed by characters of certain series of algebras. Such algebras are also constructed by J. Birman and H. Wenzl in the paper "Braids, link polynomials and a new algebra", Trans. Amer. Math. Soc. 313 (1989), 249-273.