Asymptotic Vassiliev Invariants for Vector Fields

TL;DR

This paper investigates how Vassiliev invariants grow asymptotically for ergodic vector fields in three-dimensional space, revealing that their behavior is governed by the field's helicity and applying this to classical knot invariants.

Contribution

It establishes a direct link between the asymptotics of Vassiliev invariants and the helicity of vector fields, providing explicit formulas for several knot invariants in this context.

Findings

Asymptotic growth of Vassiliev invariants is determined by helicity.

Derived formulas for asymptotic Alexander and Jones polynomials.

Provided a formula for the asymptotic Kontsevich integral.

Abstract

We analyse the asymptotical growth of Vassiliev invariants on non-periodic flow lines of ergodic vector fields on domains of $\R^3$. More precisely, we show that the asymptotics of Vassiliev invariants is completely determined by the helicity of the vector field. As an application, we determine the asymptotic Alexander and Jones polynomials and give a formula for the asymptotic Kontsevich integral.