  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.
