Minimal unlinking pathways as geodesics in knot polynomial space

• Published in: Communications physics Vol. 3; no. 1
• Main Authors: Liu, Xin, Ricca, Renzo L., Li, Xin-Fei
• Format: Journal Article
• Language: English
• Published: London Nature Publishing Group UK 10.08.2020; Nature Publishing Group
• Subjects: 639/705; 639/705/1041; Complexity; Condensed matter physics; Decay; Energy distribution; Geodesy; Knots; Physics; Physics and Astronomy; Polynomials; Probability; Simplification; Topology; Vortices
• ISSN: 2399-3650; 2399-3650
• DOI: 10.1038/s42005-020-00398-y

Physical knots observed in various contexts – from DNA biology to vortex dynamics and condensed matter physics – are found to undergo topological simplification through iterated recombination of knot strands following a common, qualitative pattern that bears remarkable similarities across fields. Here, by interpreting evolutionary processes as geodesic flows in a suitably defined knot polynomial space, we show that a new measure of topological complexity allows accurate quantification of the probability of decay pathways by selecting the optimal unlinking pathways. We also show that these optimal pathways are captured by a logarithmic best-fit curve related to the distribution of minimum energy states of tight knots. This preliminary approach shows great potential for establishing new relations between topological simplification pathways and energy cascade processes in nature. What makes a given knot's unlinking pathway more likely to appear in certain topological decay processes—from DNA to vortex dynamics? Here, the authors introduce a measure of topological complexity as distance in a suitably defined knot polynomial space, allowing them to quantify the probability of decay pathways as geodesic flows in this space.