We introduce a generalization of spin models by dropping the symmetry condition. The partition function of a generalized spin model on a connected oriented link diagram is invariant under Reidemeister moves of type II and III, giving an invariant for oriented links.

A q-analogue ζ q (s) of the Riemann zeta function ζ(s) was studied in [Kaneko et al. 03] via a certain q-series of two variables. We introduce in a similar way a q-analogue of the Dirichlet L-functions and make a detailed study of them, including some issues concerning the classical limit of ζ q (s) left open in [Kaneko et al. 03]. We also examine a "crystal" limit (i.e. q ↓ 0) behavior of ζ q (s). The q-trajectories of the trivial and essential zeros of ζ(s) are investigated numerically when q moves in (0, 1]. Moreover, conjectures for the crystal limit behavior of zeros of ζ q (s) are given. 2000 Mathematics Subject Classification : 11M06

We provide methods to compute the colored HOMFLY polynomials of knots and links with symmetric representations based on the linear skein theory. By using diagrammatic calculations, several formulae for the colored HOMFLY polynomials are obtained. As an application, we calculate some examples for hyperbolic knots and links, and we study a generalization of the volume conjecture by means of numerical calculations. In these examples, we observe that asymptotic behaviors of invariants seem to have relations to the volume conjecture.

In this paper, we relate Schur functions and a linear skein of annulus derived from the Homfly polynomail. Using this relations, we define topological invariants of 3-manifolds from the Homfly polynomial.

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