On the spectrum of Dehn twists in quantum Teichmuller theory

The operator realizing a Dehn twist in quantum Teichmuller theory is diagonalized and continuous spectrum is obtained. This result is in agreement with the expected spectrum of conformal weights in quantum Liouville theory at c>1. The completeness condition of the eigenvectors includes the integration measure which appeared in the representation theoretic approach to quantum Liouville theory by Ponsot and Teschner. The underlying quantum group structure is also revealed.Comment: 13 pages,8 figures,LaTeX2

A Link Invariant from Quantum Dilogarithm

The link invariant, arising from the cyclic quantum dilogarithm via the particular $R$-matrix construction, is proved to coincide with the invariant of triangulated links in $S^3$ introduced in R.M. Kashaev, Mod. Phys. Lett. A, Vol.9 No.40 (1994) 3757. The obtained invariant, like Alexander-Conway polynomial, vanishes on disjoint union of links. The $R$-matrix can be considered as the cyclic analog of the universal $R$-matrix associated with $U_q(sl(2))$ algebra.Comment: 10 pages, LaTe

Delta-groupoids and ideal triangulations

A Delta-groupoid is an algebraic structure which axiomatizes the combinatorics of a truncated tetrahedron. By considering two simplest examples coming from knot theory, we illustrate how can one associate a Delta-groupoid to an ideal triangulation of a three-manifold. We also describe in detail the rings associated with the Delta-groupoids of these examples.Comment: 15 pages, submitted to proceedings of the Chern-Simons gauge theory conference held in Bonn 200

Hyperbolic Structure Arising from a Knot Invariant

We study the knot invariant based on the quantum dilogarithm function. This invariant can be regarded as a non-compact analogue of Kashaev's invariant, or the colored Jones invariant, and is defined by an integral form. The 3-dimensional picture of our invariant originates from the pentagon identity of the quantum dilogarithm function, and we show that the hyperbolicity consistency conditions in gluing polyhedra arise naturally in the classical limit as the saddle point equation of our invariant.Comment: 30 pages, 18 figure

R-matrix knot invariants and triangulations

The construction of quantum knot invariants from solutions of the Yang--Baxter equation (R-matrices) is reviewed with the emphasis on a class of R-matrices admitting an interpretation in intrinsically three-dimensional terms.Comment: 13 pages, submitted to proceedings of the conference "Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory", June 2009, New Yor