| Summary: | This thesis is divided into two parts, each summarising one of the main projects I have undertaken since the beginning of my DPhil. In the first project, we study the graph associated to a knot K, whose vertices are diagrams representing K, and edges represent single Reidemeister moves. We prove that the isomorphism type of this graph is a complete knot invariant, up to mirroring. This framework suggests a discretised model to study the action of some enzymes on knotted DNA molecules. More specifically, we use a grid diagrams based model to investigate the topological consequences of intersegmental passages occurring in
circular DNA molecules. We suggest a grid diagrams-based calculation as a new and computationally convenient framework for investigating knotting probabilities in biopolymers.
Recent studies classify the topology of proteins by analysing the distribution of their projections using knotoids. In the second project, using a double branched cover construction, we prove a correspondence between knotoids and strongly invertible knots. This correspondence allows us to study knotoids through tools and invariants coming from
knot theory. By using the theory we developed, we investigate the toplogical relation between knotoids differing by small perturbations on the direction of projection. We then apply our results to infer information on the global topology of knotted proteins.
|
|---|