Knot polynomial invariants in terms of helicity for tackling topology of fluid knots
Seminar Room 1, Newton Institute
A new method based on the derivation of the Jones polynomial invariant of knot theory to tackle and quantify structural complexity of vortex filaments in ideal fluids is presented. First, we show that the topology of a vortex tangle made by knots and links can be described by means of the Jones polynomial expressed in terms of kinetic helicity. Then, for the sake of illustration, explicit calculations of the Jones polynomial for the left-handed and right-handed trefoil knot and for the Whitehead link via the figure-of-eight knot are considered. The resulting polynomials are thus function of the topology of the knot type and vortex circulation and we provide several examples of those. While this approach extends the use of helicity in terms of linking numbers to the much richer context of knot polynomials, it offers also new tools to investigate topological aspects of mathematical fluid dynamics and, by directly implementing them, to perform new real-time numerical diagnostics of complex flows.