On Alexander-Conway polynomials of two-bridge links

TL;DR

This paper studies Alexander-Conway polynomials of two-bridge links using Euler continuant polynomials, providing new proofs of classical theorems, congruences, and bounds for coefficients and roots, advancing understanding of these link invariants.

Contribution

It introduces elementary proofs of classical theorems, establishes new congruences, and derives sharp bounds for coefficients and roots of Alexander polynomials of two-bridge links.

Findings

New elementary proofs of Murasugi's and Hartley's theorems

A modulo 2 congruence for links implying a classical knot congruence

Sharp bounds for coefficients and roots of Alexander polynomials

Abstract

We consider Conway polynomials of two-bridge links as Euler continuant polynomials. As a consequence, we obtain new and elementary proofs of classical Murasugi's 1958 alternating theorem and Hartley's 1979 trapezoidal theorem. We give a modulo 2 congruence for links, which implies the classical Murasugi's 1971 congruence for knots. We also give sharp bounds for the coefficients of Euler continuants and deduce bounds for the Alexander polynomials of two-bridge links. These bounds improve and generalize those of Nakanishi Suketa'96. We easily obtain some bounds for the roots of the Alexander polynomials of two-bridge links. This is a partial answer to Hoste's conjecture on the roots of Alexander polynomials of alternating knots.