Derivation of the total twist from Chern-Simons theory

TL;DR

This paper derives the total twist, a key knot invariant, from Chern-Simons theory by connecting Wilson loop expectations to knot diagram evaluations, providing a new non-recursive formulation.

Contribution

It introduces a novel non-recursive method to compute the total twist from Chern-Simons theory, linking it to knot diagram deformations.

Findings

Derived the total twist from second order Wilson loop expressions.

Established a non-recursive evaluation method for the total twist.

Clarified the relation between the derived total twist and its original definition.

Abstract

The total twist number, which represents the first non-trivial Vassiliev knot invariant, is derived from the second order expression of the Wilson loop expectation value in the Chern-Simons theory. Using the well-known fact that the analytical expression is an invariant, a non-recursive formulation of the total twist based on the evaluation of knot diagrams is constructed by an appropriate deformation of the knot line in the three-dimensional Euclidian space. The relation to the original definition of the total twist is elucidated.