On zeros of the Alexander polynomial of an alternating knot

TL;DR

This paper investigates the zeros of the Alexander polynomial for two-bridge knots, establishing bounds on their real parts and providing new insights into their distribution.

Contribution

It proves bounds on the real parts of zeros of Alexander polynomials for two-bridge knots, including a broad class with tighter bounds.

Findings

Zeros of Alexander polynomials have real parts between -3 and 6.

For many two-bridge knots, zeros have real parts between -1 and 1.

The results improve understanding of the zero distribution of Alexander polynomials.

Abstract

We prove that for any zero {\alpha} of the Alexander polynomial of a two-bridge knot, -3 < Re({\alpha}) < 6. Furthermore, for a large class of two-bridge knots we prove -1<Re({\alpha}).