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⩽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→∞, 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 3 n 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 ±=e ±2 πi/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→∞.