Orateur: Renzo Ricca (Milan)
Titre: Vortex knots cascade measured by HOMFLYPT polynomial
Résumé:  Topological fluid mechanics has a long history, experiencing a recent revival since Moffatt’s original work of 1969 through the topological interpretation of kinetic helicity of vortex dynamics, one of the most fundamental invariants of ideal fluid flows, in terms of the Gauss linking number. For a single vortex filament Moffatt and Ricca (1992) showed that helicity can be expressed in terms of the Calugareanu-White self linking number in terms of writhe and twist of the vortex knot. Recently, by applying knot theoretical techniques Liu and Ricca have derived well-known knot polynomials [1, 2] – most notably the HOMFLYPT polynomial– from the helicity of fluid systems, hence showing that these can provide indeed new invariants of ideal fluid mechanics, and in the case of HOMFLYPT the two polynomial variables have been shown to be related to the writhe and twist of the vortex knot.

Due to reconnection or recombination of neighboring strands vortex knots can undergo an almost generic cascade process, that tend to reduce topological complexity by stepwise unlinking. Here, by using the adapted HOMFLYPT polynomial for fluid knots, we prove that under the assumption that topological complexity decreases by stepwise unlinking by anti-parallel reconnection, this cascade process follows a path detected by a unique, monotonically decreasing sequence of numerical values [3]. This result holds true for any sequence of standardly embedded torus knots T(2, 2n + 1) and torus links T(2, 2n). By this result we demonstrate that the computation of this adapted HOMFLYPT polynomial provides a powerful tool to quantify topological complexity of various physical systems.

---