Operator Formalism for Topology-Conserving Crossing Dynamics in Planar Knot Diagrams

Abstract

We address here the topological equivalence of knots through the so-called Reidemeister moves. These topology-conserving manipulations are recast into dynamical rules on the crossings of knot diagrams. This is presented in terms of a simple graphical representation related to the Gauss code of knots. Drawing on techniques for reaction-diffusion systems, we then develop didactically an operator formalism wherein these rules for crossing dynamics are encoded. {The aim is to develop new tools for studying dynamical behaviour and regimes in the presence of topology conservation}. This necessitates the introduction of composite paulionic operators. The formalism is applied to calculate some differential equations for {the time evolution} of densities and correlators of crossings, subject to topology-conserving stochastic dynamics. {We consider here the simplified situation of two-dimensional knot projections. However, we hope that this is a first valuable step towards} addressing a number of important questions regarding the role of topological constraints {and specifically of topology conservation} in dynamics through a variety of solution and approximation schemes. Further applicability arises in the context of the simulated annealing of knots. The methods presented here depart significantly from the invariant-based path integral descriptions often applied in polymer systems, {and, in our view, offer a fresh perspective on} the conservation of topological states and topological equivalence in knots.