Tackling structural complexity by Jones' polynomials
Seminar Room 1, Newton Institute
In this paper we present new results based on applications of knot theory to tackle and quantify structural complexity of vortex dynamics. In the ideal case of Euler equations we show that the topology of any vortex tangle, made by knots and links, can be identified and described by the Jonesís polynomials of the tangle, expressed in terms of kinetic helicity. Explicit calculations of the Jones polynomial for the left-handed and right-handed trefoil knots and for the Whitehead link via the figure-of-eight knot are worked out for illustration. This novel approach contributes to establish a fundamentally new paradigm in topological fluid mechanics, by extending the former interpretation of helicity in terms of linking numbers to the much richer context of knot polynomials, and by opening up new directions of work both in mathematical fluid dynamics and in numerical diagnostics of vortex flows.