The Vassiliev invariant of degree 2 as a tool to study protein structure

Abstract

Knot theory refers to the mathematical study of the objects with the same name we encounter in daily life. Its main goal is to determine when two knots are equivalent. Knot invariants are tools that allow us to determine this. Results coming from knot theory have been useful to other areas of knowledge, such as chemistry, fluid dynamics and molecular biology. In my thesis, I present one of these applications to the study of protein structure. For this purpose, I used two knot invariants for knots: the Alexander polynomial and the Vassiliev invariant of degree 2. This is the subject of Chapter 1. I present their generalization to objects called "tangles" in Chapter 2. The tool that provides this generalization is the algebraic structure called "meta-monoid". In Chapter 3 I include how to construct a graph of a protein from information coming from the Protein Data Bank (PDB) online repository. Also, I explain how to derive a simplified structure from which we can get a tangle. The computation of the Vassiliev invariant of degree 2 of the protein with PDB entry 1L2Y appears in Chapter 4. The code in Python that translates the information of the PDB files into a tangle (with a graph as intermediate step) appears in Chapter 5.