From Knot Invariants to Knot Dynamics

This paper is an introduction to combinatorial topology via state summation models for the Jones polynomial and its generalizations and also an introduction to specific aspects of the dynamics of knots as they are influenced by force fields, and how knotted vortices undergo reconnection in their process of simplification. We consider the world line of such a knotted vortex and show how the relationships of knot theory and four dimensional topology can be used to give bounds on its reconnection number and an exact formula for the reconnection number in the case of positive knots, including all torus knots.