New routes to quantify topological complexity by adapted polynomials

Renzo Ricca (Universita' Milano Bicocca)

Thursday 20th February, 2020 14:00-15:00 Maths 311B

Abstract

Filamentary structures are ubiquitous in nature, data sets and technology. Topological characterization of complex entanglements in physical systems and networks is as useful in science as it is in applications to detect and predict critical phenomena [1]. Here we present new results based on applications of knot theoretical concepts to quantify topological entanglement of filamentary structures in complex systems. For the sake of example we consider the cascade process of fluid knots under natural reconnection and recombination and we show how standard knot polynomials such as Jones and HOMFLYPT can be adapted to quantify topology. This is done by associating writhe and twist of fluid knots to the polynomial variables. We show that our adapted polynomials provide useful, quantitative measurements of topological states and transitions of the fluid structures [2]. Under certain general assumptions we find that generic cascade processes are actually detected by a unique, monotonic decreasing sequence of numerical values [3]. By comparing various knot polynomials, we show that HOMFLYPT proves to be a robust marker for numerical diagnostics in the analysis of big data sets [4] and show how these can be readily implemented in the diagnostics of superfluid turbulence.

References

[1]     Ricca, R.L. (2005) Structural complexity. In Encyclopedia of Nonlinear Science (ed. A. Scott), pp. 885-887. Routledge, New York and London.

[2]     Liu, X. & Ricca, R.L. (2015) On the derivation of HOMFLYPT polynomial invariant for fluid knots. J. Fluid Mech. 773, 34-48.

[3]     Liu, X. & Ricca, R.L. (2016) Knots cascade detected by a monotonically decreasing sequence of values. Nature Sci. Rep. 6, 24118.

[4]     Ricca, R.L. & Liu, X. (2018) HOMFLYPT polynomial is the best quantifier for topological cascades of vortex knots. Fluid Dyn. Res. 50, 011404.      

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