The Jones polynomial for fluid knots from helicity

Abstract

In this paper we prove that under ideal conditions the helicity of fluid knots, such as vortex filaments or magnetic flux tubes, provides a fundamentally new topological means by which we may associate a topological invariant, the Jones polynomial, that is much stronger than prior interpretations in terms of Gauss linking numbers. Our proof is based on an extension of the Kauffman bracket polynomial for unoriented knots. Explicit calculations of the Jones polynomial for the left- and right-handed trefoil knots and for the Whitehead link via the figure-of-eight knot are presented for illustration. This novel approach establishes a topological foundation of classical field theory in general, and of mathematical fluid dynamics in particular, by opening up new directions of work both in theory and applications. © 2012 IOP Publishing Ltd.