Zeroes of the Jones polynomial

Abstract

We study the distribution of zeroes of the Jones polynomial $V_K(t)$ for a knot $K$. We have computed numerically the roots of the Jones polynomial for all prime knots with $N\leq 10$ crossings, and found the zeroes scattered about the unit circle $|t|=1$ with the average distance to the circle approaching a nonzero value as $N$ increases. For torus knots of the type $(m,n)$ we show that all zeroes lie on the unit circle with a uniform density in the limit of either $m$ or $n\to \infty$, a fact confirmed by our numerical findings. We have also elucidated the relation connecting the Jones polynomial with the Potts model, and used this relation to derive the Jones polynomial for a repeating chain knot with $3n$ crossings for general $n$. It is found that zeroes of its Jones polynomial lie on three closed curves centered about the points $1, i$ and $-i$. In addition, there are two isolated zeroes located one each near the points $t_\pm = e^{\pm 2\pi i/3}$ at a distance of the order of $3^{-(n+2)/2}$. Closed-form expressions are deduced for the closed curves in the limit of $n\to \infty$.