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Arithmetic invariants of torus links

Anwesh Ray, Tanushree Shah

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 $Δ_{p,q}(t)$, 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 $Δ_{p,q}(t)$ vanishes identically and the associated homological growth in towers of abelian covers of $S^3$ branched along $T_{p,q}$ is subexponential.

Arithmetic invariants of torus links

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 and analyze the asymptotic behaviour of the zeros of their Alexander polynomials , defined with respect to the total linking number covering. We prove that as these zeros become equidistributed on the unit circle and derive an explicit formula for the limiting frequency with which primitive -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 vanishes identically and the associated homological growth in towers of abelian covers of branched along is subexponential.

Paper Structure

This paper contains 22 sections, 32 theorems, 210 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. Main results
  4. Outlook
  5. Preliminary notions
  6. Torus knots
  7. Alexander polynomials and coloring invariants
  8. The double branched cover
  9. Coloring torus knots
  10. Distribution of zeros of Alexander polynomials
  11. Distribution results for torus knots
  12. Distribution results for the family of torus links
  13. Weak convergence of measures
  14. Multiplicity statistics for torus knots
  15. Moments of roots of torus knots and their generating function
  16. ...and 7 more sections

Key Result

Theorem A

Let $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$ and write each $z\in\mathbb{T}$ as $z=e^{2\pi i\theta(z)}$ with $\theta(z)\in[0,1)$. For any interval $[a,b]\subset[0,1]$,

Figures (2)

  • Figure 1: Trefoil: torus (2,3) knot
  • Figure 2: Cinquefoil: torus (2,5) knot

Theorems & Definitions (60)

  • Theorem A: Theorem \ref{['section 3 main thm 1']}
  • Theorem B: Theorem \ref{['section 3 main thm 2']}
  • Theorem C: Theorem \ref{['measure theoretic convergence theorem']}
  • Theorem D: Theorem \ref{['thm:avg-Mr']}
  • Theorem E: Theorem \ref{['Theorem G p q residues']}
  • Theorem F: Corollary \ref{['cor:torus-measure-zero']} and Corollary \ref{['cor 4.8']}
  • Theorem G: Theorems \ref{['thm 5.3']} and \ref{['thm 5.4']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • ...and 50 more