Conway polynomials of two-bridge links

TL;DR

This paper establishes necessary conditions for Conway polynomials of two-bridge links, providing new proofs of classical theorems, congruences, and improved bounds for polynomial coefficients.

Contribution

It introduces new necessary conditions and bounds for Conway and Alexander polynomials of two-bridge links, enhancing understanding and generalizing previous results.

Findings

Necessary conditions for Conway polynomials of two-bridge links

Simplified proofs of Murasugi and Hartley's theorems

Sharp bounds for polynomial coefficients

Abstract

We give necessary conditions for a polynomial to be the Conway polynomial of a two-bridge link. As a consequence, we obtain simple proofs of the classical theorems of Murasugi and Hartley. We give a modulo 2 congruence for links, which implies the classical modulo 2 Murasugi congruence for knots. We also give sharp bounds for the coefficients of the Conway and Alexander polynomials of a two-bridge link. These bounds improve and generalize those of Nakanishi and Suketa.