Minimal unlinking pathways as geodesics in polynomial space

Monday, June 24, 2019 - 2:30pm - 3:30pm
Keller 3-180
Renzo Ricca (Università di Milano - Bicocca)
Complex fluid structures, such as linked and knotted vortex tubes in classical or quantum systems, undergo a series of reconnection events that lead to a cascade process through successive topological simplifications. To a certain extent the process is identical to the DNA unlinking scenario studied in recombinant DNA plasmids [1]. A confirmation of this comes also from the application of adapted knot polynomials, such as Jones and HOMFLYPT, to the cascade process of torus knots and links, where we showed [2] that indeed the process was governed by a sequence of monotonically decreasing HOMFLYPT numerical values. Here we propose a new theoretical framework where the minimal unlinking pathways analyzed in [1] are recovered as geodesics in an appropriate knot polynomial space. For simplicity we consider Jones polynomials and a flat metric given by the inner product of Legendre polynomials. Based on this a new definition of topological complexity is introduced and it is shown to provide useful information for optimal unlinking processes. Preliminary computations show a very good agreement with the results obtained in [1].

This is joint work with Xin LIU and Xinfei LI (BJUT, P.R. China).
[1] Stolz, R., Yoshida, M., Brasher, R., Flanner, M., Ishihara, K., Sherratt, D.J., Shimokawa, K. & Vazquez, M. (2017) Pathways of DNA unlinking: A story of stepwise simplification. Nature Sci. Rep. 7, 12420.
[2] Liu, X. & Ricca, R.L. (2016) Knots cascade detected by a monotonically decreasing sequence of values. Nature Sci. Rep. 6, 24118.