Quantifying the hierarchical topological decay of fluid knots

In this paper we analyze the hierarchical topological cascade of knotted and linked filaments associated with the natural decay of superfluid vortex tangles. This is done by using an algebraic reformulation of knot polynomial invariants to quantify the topological states of the cascade process. By interpreting the evolutionary paths associated with vortex knot dynamics as geodesic flows in an appropriate knot polynomial space defined by adapted Jones polynomials [X. Liu, R. L. Ricca, J. Phys. A Math. Theor. 45 , 205501 (2012).], we introduce an entanglement index to quantify the probability density of occurrence of each topological state. By this method we analyze all prime knots, links, and composites up to 8 crossings, and show that this measure detects accurately the decreasing topological complexity observed in simulations and laboratory experiments. By a comparative analysis carried out with data based on an alternative measure (complexity degree), we show that this entanglement index proves to be a more robust and reliable quantifier. The theoretical framework proposed here can be applied to other types of physical systems, where networks of filamentary structures are subject to continuous topological change and reorganization.