A computational library for knotted structures and applications

Proteins are large complex molecules made up of chains of amino acids. They are essential to life and practically support all its functions. Here knot theory plays an important role, as knots have been identified in both DNA and proteins. Kinetic difficulties in protein knot formation might suggest that nature is biased against forming knots during evolutionary processes, but this is certainly not the case. It is argued that knottedness provides structural advantages of the molecule, for example, protection against degradation, thermal and kinetic stability. In recent years, extensive research has been made showing that knots in proteins form an important structure layer that captures both the functional and structural role of proteins (and DNA). However, it is difficult to describe this (topological) layer, and several models have been proposed in literature so far: knots, knotoids, virtual knots, theta-curves, bonded knots, bondoids, etc. The goal of the proposed 3-year research project is to develop a robust and original computational library for computing with several kinds of knotted structures with real-world applications. We will develop a Python library containing the main PlanarDiagram class that will provide a common framework for multiple types of knotted structures. Such a data structure will enable us to encode a wide variety of knotted structures, such as knots, knotoids, virtual knots, spatial graphs, … The main functionalities of the library will be the following: structure manipulation tools (Reidemeisiter moves), knot detection tools, knot identification tools (identifying the knotted isotopy class of the structure using canonization and minimalization of diagrams), computation of invariants (multivariable Alexander polynomial, Jones polynomial, HOMFLYPT polynomial, Yamada polynomial,…) and several visualization tools. The PlanarDiagram class will be flexible and extendible in the sense that the user will be able to define a custom set of local moves and pass them into the object. We will also develop tables of knotted structures for bondoids and bonded virtual knots. We will analyse the proteins from the Protein Data Bank (consisting of nearly 200,000 entries) by classifying them by their bondoid type and bonded virtual knot type. Furthermore, a statistical analysis, based on this classification, will be made. We will also demonstrate that the library can also be used to study knots in closed, connected, orientable 3-manifold. Such knots can be easily presented as a PlanarDiagram object in form of a mixed link diagram (a diagram consisting of the regular knot projection inside the Kirby diagram of the 3-manifold). We will implement invariants for such knots (the Alexander polynomial and the Kauffman bracket skein module). We expect our library to be one of the central libraries to study knot theory, knotted biomolecules and knots in 3-manifolds. Understanding protein entanglement is crucial to determining the structural, functional, and evolutionary role of topology in proteins and DNA. We believe that this project, which combines topics from topology, combinatorics, molecular biology, bioinformatics, and computer science, will make great and interesting advancements in all these fields.