Unity of Jones polynomials in the unit circle and the plane

TL;DR

This paper investigates the solutions of the Jones polynomial equation $J_K(t)=1$, demonstrating that roots of unity and complex zeros are densely populated in the unit circle and the plane for specific knot and link families.

Contribution

It establishes the density of solutions to $J_K(t)=1$ for double-twist knots and constructs link families with zeros dense in the complex plane.

Findings

Roots of unity (except -1) satisfy $J_{K_n}(ta)=1$ for some $n$.

Solutions to $J_{K_n}(t)=1$ are dense in the unit circle.

Zeros of $J_L(t)-1$ are dense in the complex plane for certain link families.

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 $J_{K_n}(\zeta)=1$ 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.