Abstract
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.
Highlights
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
Since we refer to physical knots, we make use of the adapted polynomials introduced by Liu and Ricca[18,19] to quantify the topology of fluid knots.This is done by taking standard Legendre polynomials to construct a basis and an appropriate metric to compute distance using knot polynomials
We find that the data obtained by our approach match very well the data obtained by the other method, suggesting that the alternative method based on use of geodesics in knot polynomial space has great potential for probability computation of decaying processes of physical systems, even far more complex than the one considered here
Summary
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. This allows to measure and compare geodesic distances between knot types, define and compute relative probabilities, and relate minimal pathways to lower bounds given by minimum energy states[20,21,22].
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