Arithmetic invariants of torus links

TL;DR

This paper investigates the asymptotic distribution of zeros of Alexander polynomials of torus links, revealing equidistribution on the unit circle and analyzing their statistical and algebraic properties.

Contribution

It provides a detailed analysis of the zero distribution, introduces the moment sequence, and explores Iwasawa invariants for torus links, offering new insights into their arithmetic invariants.

Findings

Zeros become equidistributed on the unit circle as p,q grow large

Explicit formula for the limiting frequency of primitive roots of unity

Logarithmic Mahler measure vanishes, and homological growth is subexponential

Abstract

The classical analogy between knots and primes motivates the study of Alexander polynomials through an arithmetic perspective. In this article we study the two-parameter family of torus knots and links $T_{p,q}$ and analyze the asymptotic behaviour of the zeros of their Alexander polynomials $\Delta_{p,q}(t)$, defined with respect to the total linking number covering. We prove that as $p,q\to\infty$ these zeros become equidistributed on the unit circle and derive an explicit formula for the limiting frequency with which primitive $r$-th roots of unity appear. To capture finer statistical information, we introduce the moment sequence of the zero distribution and compute its generating function in closed form. We further examine the Iwasawa theory of the corresponding branched covers, determining the Iwasawa invariants. The logarithmic Mahler measure of $\Delta_{p,q}(t)$ vanishes identically and the associated homological growth in towers of abelian covers of $S^3$ branched along $T_{p,q}$ is subexponential.