A topological approach to vortex knots and links

Here we review some recent developments in the topological study of vortex knots and links. We start from fundamental properties of kinetic helicity, a conserved quantity of ideal fluid mechanics, focusing on the topological interpretation in terms of linking numbers. Then we proceed to consider the derivation from helicity of the Jones and HOMFLYPT knot polynomials, showing that their adapted formulation can be expressed in terms of writhe and twist helicity. Through explicit computations and examples, we also show that these adapted polynomials provide useful information to quantify the topological cascade of reconnecting vortex knots and links in relation to energy. In an effort to establish direct relationships between topological complexity and energy contents, we highlight the role of ropelength in the energy relaxation of magnetic knots. Optimal pathways of decaying vortex knots are re-considered by a new approach based on the interpretation of geodesic flows in an appropriate knot polynomial space. This novel approach proves to be useful to determine the probability associated with each topological state, and the topological degeneration of fluid structures undergoing a change of topology. As an example, we consider the numerical simulation of the decay of 3 vortex loops linked together to form a system of Borromean rings. The evolution of such system is governed by the Gross-Pitaevskii equation, and preliminary results of this test case are analysed in terms of energy transfers.