Quantum Morphing and the Jones Polynomial

Abstract

We will explore the experimental observation that on the set of knots with bounded crossing number, algebraically independent Vassiliev invariants become correlated, as noticed first by S. Willerton. We will see this through the value distribution of the Jones polynomial at roots of unit. As the degree of the roots of unit is getting larger, the higher order fluctuation is diminishing and a more organized shape will emerge from a rather random value distribution of the Jones polynomial. We call such a phenomenon "quantum morphing". Evaluations of the Jones polynomial at roots of unity play a crucial role, for example in the volume conjecture. When

Models of random knots

“…Besides representing the first two finite type invariants of knots, the planar map u : K7 !ðc 2 ðKÞ; v 3 ðKÞÞ has some interesting properties. As observed by Dasbach et al (2001), the evaluation of the Jones polynomial at roots of unity near 1 can be approximated by V K ðe ih Þ ¼ 1 þ 3c 2 h 2 þ 6v 3 h 3 i þ Oðh 4 Þ, and this yields similar fish graphs for V K ðe 2pi=N Þ in the complex plane, for N ) n. Note that by the multiplicativity of the Jones polynomial, the map u is additive with respect to connected sum: uðK#K 0 Þ ¼ uðKÞ þ uðK 0 Þ in Z 2 . Using this fact and some known constructions one can show that as n grows the resulting point set of all ðc 2 =n 2 ; v 3 =n 3 Þ is dense in R 2 .…”