Our recent paper (See post here; CM Rohwer, KK Müller-Nedebock “Operator Formalism for Topology-Conserving Crossing Dynamics in Planar Knot Diagrams”, Journal of Statistical Physics 159, 120-157 (2015), http://dx.doi.org/10.1007/s10955-014-1161-1) introduces a dynamical perspective to the venerable polymer physics problem of conserving the state of entanglement in polymer loops.
To understand what this means we show the example of a single strand (polymer) that forms a closed loop, but, as in the accompanying figure of the so-called trefoil knot, cannot be manipulated into a simple circular form without cutting and re-glueing the strand. We can now ask how one can characterise the strand as it is subjected to continuous changes of its configurations due to the environment in which it finds itself. (Think of a polymer loop in a fluid.) As it reconfigures its shape changes, but the topology of the knot must remain invariant. The statistical physics entails computing the sum over all possible configurations maintaining the topology. Great papers by Edwards and Frisch and Wasserman have done this elegantly. Yet many severe and interesting mathematical challenges still face the enumeration of conformations, but also just in checking whether two knots are equivalent in a way that can be suitably implemented in statistical physics.
Our dynamical formalism is one possible approach in which one can start thinking about a dynamical route to exploring the possible permissible configurations (albeit in the reduced two-dimensional crossings in the projection of the knot only) without needing to compute and compare knot invariants.
Previously, we also published an alternative enumeration scheme to knots in which the winding number is preserved (See the related post; CM Rohwer, KK Müller-Nedebock, F-E Mpiana Mulamba “Conservation of polymer winding states: a combinatoric approach” J Phys A-Math Theor 47, 065001 (2014) DOI: http://dx.doi.org/10.1088/1751-8113/47/6/065001)