New method for computation of fluid helicity: Knot polynomial invariants

TL;DR

This paper introduces an algebraic approach linking fluid helicity to knot polynomial invariants, enabling computation via skein relations and recursion, thus bridging fluid dynamics and knot theory.

Contribution

It develops a novel algebraic method that relates fluid helicity to knot polynomial invariants using skein relations, providing a new computational framework.

Findings

Established a topological invariant related to fluid helicity

Demonstrated skein relations satisfy Kauffman polynomial properties

Enabled recursive computation of helicity through algebraic invariants

Abstract

A new algebraic method for computing helicity is developed, by discovering a relationship between helicity of fluid mechanics and algebraic polynomial invariants of knot theory. We have constructed a topological invariant $t^{H\left(\mathcal{L}\right)}$ for a link $\mathcal{L}$ of knots, where $H$ is the helicity of a given fluid and $t$ a formal constant. For oriented knotted vortex lines, $t^{H}$ satisfies the skein relations of the Kauffman R-polynomial; for un-oriented knotted lines, $t^{H}$ satisfies the skein relations of the Kauffman bracket polynomial. Our new algebraic method is to use skein relations to compute the helicity of a link $\mathcal{L}$ by algebraic recursion.