Field-line helicity and adapted polynomials in vortex knots dynamics

<p>The spontaneous decay of tangles of vortex filaments in a turbulent fluid can be described as a topological simplification process that brings a complex network of vortex knots and links to simplify by a finite sequence of reconnections towards an ensemble of unknotted unlinked loops. Here we discuss the key topological stages of this process, and show that a generic vortex cascade process can be analysed by the application of adapted knot polynomials derived from the field-line helicity of fluid mechanics. Since this quantity plays a crucial role in the computation of the adapted polynomials, we discuss some of its key properties in connection with the Wilson loop operator of field theory. To illustrate the use of these polynomials we refer to the simulated time evolution of three interlinked vortices forming Borromean rings and compare the simulation results with the numerical values provided by the polynomial computation of the simplifying pathways interpreted as geodesic flows in an appropriately defined knot polynomial space.</p>