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.