Braid zeta function and some formulae for the torus type

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 can construct a new zeta function of an element of the braid group.In this paper,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 define the function $Z_q$ associated with some braids. We show that this function can be expressed by some braid zeta function for the case of special braids whose closures are isotopic to certain torus knots.