Zeroes of the Jones polynomial

Details Zeroes of the Jones polynomial Zeroes of the Jones polynomial

2001-05-01

arxiv.org/abs/cond-mat/0105013v2

Physica A 296, 483-494 (2001).

Physics

Condensed Matter

Statistical Mechanics

12 pages, 5 figures

Scientific paper

10.1016/S0378-4371(01)00189-3

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 $n\to \infty$.

Affiliated with

Also associated with

No associations

Rating

Zeroes of the Jones polynomial does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Zeroes of the Jones polynomial, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Zeroes of the Jones polynomial will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-5251