Braid zeta function and some formulae for the torus type

TL;DR

This paper introduces a new braid zeta function inspired by a known zeta function of symmetric group elements, relating it to the Alexander polynomial and torus knots, offering novel formulas and connections in knot theory.

Contribution

It constructs a new braid zeta function and demonstrates its relation to the Alexander polynomial and torus knots, providing new formulas in knot invariants.

Findings

Alexander polynomial expressed via braid zeta function

Defined $Z_q$ function related to braid zeta functions

Expressed $Z_q$ for torus knots as braid zeta functions

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 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.