Unity of Jones polynomials in the unit circle and the plane

Michal Jablonowski

paper

Unity of Jones polynomials in the unit circle and the plane

arXiv:2603.14585

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.