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Zeroes of the Jones polynomial

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  • Wu, F.Y
  • Wang, J

Abstract

We study the distribution of zeroes of the Jones polynomial VK(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 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±=e±2πi/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→∞.

Suggested Citation

  • Wu, F.Y & Wang, J, 2001. "Zeroes of the Jones polynomial," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 296(3), pages 483-494.
    • Handle: RePEc:eee:phsmap:v:296:y:2001:i:3:p:483-494
    DOI: 10.1016/S0378-4371(01)00189-3
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    References listed on IDEAS

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    1. Shrock, Robert, 2000. "Exact Potts model partition functions on ladder graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 283(3), pages 388-446.
    2. Chang, Shu-Chiuan & Shrock, Robert, 2000. "Exact Potts model partition function on strips of the triangular lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 286(1), pages 189-238.
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