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Unity of Jones polynomials in the unit circle and the plane

Michal Jablonowski

Abstract

In this note, we study solutions of the equation $J_K(t)=1$ for the Jones polynomial of knots and links. For the family $K_n$ of double-twist knots, we show that every root of unity (except $-1$) satisfies $J_{K_n}(ζ)=1$ for some $n$. Consequently, the set of solutions to $J_{K_n}(t)=1$ arising from this family is dense in the unit circle. We further show that there exists a family of links for which the zeros of $J_L(t)-1$ are dense in the complex plane, adapting the density mechanism of Jin--Zhang--Dong--Tay for Jones polynomial zeros.

Unity of Jones polynomials in the unit circle and the plane

Abstract

In this note, we study solutions of the equation for the Jones polynomial of knots and links. For the family of double-twist knots, we show that every root of unity (except ) satisfies for some . Consequently, the set of solutions to arising from this family is dense in the unit circle. We further show that there exists a family of links for which the zeros of are dense in the complex plane, adapting the density mechanism of Jin--Zhang--Dong--Tay for Jones polynomial zeros.
Paper Structure (3 sections, 3 theorems, 25 equations, 1 figure)

This paper contains 3 sections, 3 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Density in the unit circle
  3. Density in the complex plane

Key Result

Theorem 2.1

Let $J_n(t)$ denote the Jones polynomial of $K_n$ normalized so that $J_n(1)=1$. Then $\left\{t\in S^1:\exists n\ge1,\ J_n(t)=1\right\}$ is dense in the unit circle $S^1$. More precisely, where $\mu_\infty$ denotes the set of all roots of unity.

Figures (1)

  • Figure 1: Solutions to $J(t)=1$ for prime knots up to $16$ crossings.

Theorems & Definitions (5)

  • Theorem 2.1
  • proof
  • Proposition 3.1
  • Lemma 3.2: Beraha--Kahane--Weiss BKW78
  • proof : Proof of Proposition \ref{['prop:BKW3']}