Knots and knot representations in chemistry: a graph theory approach

Abstract

In the following thesis graph theory and knot theory are introduced using chemistry related problems and how these problems can be solved using these theories. This is followed by the description of knot projection method and its importance in defining and describing knot representations and their use as similarity measures. The first chapter focuses on graph theory, its foundation, some graph characteristics, graph colouring, some special graphs and graph matrices. This is followed by an example problem that can be solved using graph theory. Chapter Two introduces the basics of knot theory, some topological terminology, the Reidemeister moves, symmetry and chirality, the basics of the most popular knot polynomials and the skein relation. The third chapter is dedicated to the structure of proteins from primary to quinary. Chapter Four presents the research starting with knot projections, where graph theory meets knot theory. Projecting proteins on a plane and orientation problems, then possible solutions and simplifications to these problems such as projecting on a sphere. The knot projection graph, its polynomial and interesting properties related to chirality, symmetry and similarity measures are shown. Conclusions are discussed in Chapter Five with possible uses of knot projections and plans on future work. The code that was written for this project can be used freely and it is attached in the appendix.