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.

No abstract

We calculate the cosmological complexity under the framework of scalar curvature perturbations for a K-essence model with constant potential. In particular, the squeezed quantum states are defined by acting a two-mode squeezed operator which is characterized by squeezing parameters r k and φ k on vacuum state. The evolution of these squeezing parameters are governed by the Schrödinger equation, in which the Hamiltonian operator is derived from the cosmological perturbative action. With aid of the solutions of r k and φ k , one can calculate the quantum circuit complexity between unsqueezed vacuum state and squeezed quantum states via the wave-function approach. One advantage of K-essence is that it allows us to explore the effects of varied sound speeds on evolution of cosmological complexity. Besides, this model also provides a way for us to distinguish the different cosmological phases by extracting some basic informations, like the scrambling time and Lyapunov exponent etc, from the evolution of cosmological complexity.

No abstract