Zeta Function Associated with the Representation of the Braid Group

Abstract

There is a well-known zeta function of the $\mathbb {Z}$-dynamical system generated by an element of the symmetric group. By considering this zeta function as a model, we construct a new zeta function of an element of the braid group . In this article, we show that the Alexander polynomial which is the most classical polynomial invariant of knots can be expressed in terms of this braid zeta function. Furthermore, we show that the zeta function associated with the tensor product representation $\beta _{n, q}^{\otimes r}$ can be expressed by some braid zeta function for the case of special braids whose closures are isotopic to certain torus knots . Moreover, we introduce the zeta function associated with the Jones representation which is defined by using the R-matrix satisfying the Yang–Baxter equation. Then, we calculate this zeta function for $n=3$ and show the relation between the Alexander polynomial and the Jones polynomial.