Applications of R-graphs to DNA modelling

Finding implementable descriptions of the possible configurations of a knotted DNA molecule has remarkable importance from a biological point of view, and it is a hard and well studied problem in mathematics.
Here we present two newly developed mathematical tools that describe the configuration space of knots and model the action of Type I and II Topoisomerases on a covalently closed circular DNA molecule: the Reidemeister graphs.
We determine some local and global properties of these graphs and prove that in one case the graph-isomorphism type is a complete knot invariant up to mirroring.
Finally, we indicate how the Reidemeister graphs can be used to infer information about the proteins' action.