On volume-preserving vector fields and finite type invariants of knots

TL;DR

This paper introduces a new family of invariants for divergence-free vector fields in three-dimensional space, extending Arnold's asymptotic linking number using configuration space integrals, to connect vector field properties with finite type knot invariants.

Contribution

It develops a novel framework for associating finite type invariants of knots to divergence-free vector fields via configuration space integrals.

Findings

Defined a new family of invariants for divergence-free vector fields

Extended Arnold's asymptotic linking number using finite type invariants

Utilized configuration space integrals by Bott and Taubes

Abstract

We consider the general nonvanishing, divergence-free vector fields defined on a domain in three space and tangent to its boundary. Based on the theory of finite type invariants, we define a family of invariants for such fields, in the style of Arnold's asymptotic linking number. Our approach is based on the configuration space integrals due to Bott and Taubes.