Knot theory, braiding and circuit topology
Collaborators:
- Alireza Mashaghi (University of Leiden, The Netherlands)
- Ioannis Diamantis (University of Maastricht, The Netherlands)
A knot refers to a closed loop embedded in three-dimensional space. Knots are characterized by their topological properties, and are hence invariant under any continuous transformation. One of the primary goals of knot theory is to classify different types of knots and understand their properties. Knots are considered equivalent if they can be deformed into one another without cutting or passing the strand through itself, and it is in some cases possible to find an invariant property of knots that is able to distinguish between them (e.g., the HOMFLYPT polynomial or determinant).
Braiding theory, on the other hand, is closely related to knot theory. It deals with the study of braids, which are collections of interlaced strands or strings.The connection between knot theory and braiding theory arises from the fact that knots can be represented as closed braids. By mapping the strands of a braid onto a cylindrical surface and connecting the endpoints, one can obtain a corresponding knot. This correspondence allows for the translation of knot theory problems into the realm of braiding theory and vice versa.
Circuit topology describes relationships between contacts on a linear string, e.g., a polymer. Contacts can be in parallel (P), series (S), or cross (X) relation. Intervals define the span between contacts: nonoverlapping for series, partially overlapping for cross, and one contained within the other for parallel.
My research aims to merge the three aforementioned fields by studying how, in braided bundles of short chains, topological braiding properties such as the writhe, complexity or isotopy class are related to the circuit topology content of the system. While the former are only concerned with entenglement of the participating chains, the latter only considers hard contacts, i.e., positions on the chain that interact and can for example attract one another. By merging the two aspects, my research is able to provide a clearer picture of the topology of polymer bundles.