Strong coprimality and strong irreducibility of Alexander polynomials

TL;DR

The paper investigates strong irreducibility and coprimality of Alexander polynomials, establishing conditions and applying results to knot theory, particularly for twist knots, to analyze their algebraic and concordance properties.

Contribution

It provides new criteria for strong irreducibility and coprimality of Alexander polynomials, and applies these to understand the structure of the knot concordance group.

Findings

Alexander polynomials of twist knots are pairwise strongly coprime

Most twist knots have strongly irreducible Alexander polynomials

Explicit knots are constructed as independent elements in the solvable filtration

Abstract

A polynomial f(t) with rational coefficients is strongly irreducible if f(t^k) is irreducible for all positive integers k. Likewise, two polynomials f and g are strongly coprime if f(t^k) and g(t^l) are relatively prime for all positive integers k and l. We provide some sufficient conditions for strong irreducibility and prove that the Alexander polynomials of twist knots are pairwise strongly coprime and that most of them are strongly irreducible. We apply these results to describe the structure of the subgroup of the rational knot concordance group generated by the twist knots and to provide an explicit set of knots which represent linearly independent elements deep in the solvable filtration of the knot concordance group.