Vassiliev measures of complexity for open and closed curves in 3-space

TL;DR

This paper introduces Vassiliev measures of complexity for open and closed curves in 3-space, linking them to the enhanced Jones polynomial and providing computational methods including integral formulations and geometric probabilities.

Contribution

It defines new Vassiliev measures for open curves, relates them to the enhanced Jones polynomial, and offers practical computation techniques for these measures.

Findings

Vassiliev measures are continuous functions of curve coordinates.

For closed curves, the second Vassiliev measure can be computed via a Gauss code diagram.

The double alternating self-linking integral simplifies for polygonal curves, relating to geometric probabilities.

Abstract

In this manuscript we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates and as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant for closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities. For a polygonal curve with 4 edges, the double alternating self-linking integral coincides with the signed geometric probability of obtaining the knotoid k2.1 in a random projection direction.