Knot theory

Fig.1.    A knot on a rope                                                       Fig.2.     It is called a square knot

Knots have been extremely beneficial through the ages to our actual existence and progress. For example, in the primordial ages of our existence, in order to construct an axe, a piece of stone was bound/knotted to a sturdy piece of wood. To make a net, vines or creepers, animal hairs, et cetera were bound/braided together. Even the shoelaces and ropes appearing in our daily life are related to the concepts of knots. If you want to fasten some materials such as rope by tying or interweaving, then it is just a knotting process.

Although people have been making use of knots since the dawn of our existence, the actual mathematical study of knots is relatively young, closer to 100 years than 1000 years. In contrast, (Euclidean) geometry and properties of numbers, which have been studied over a considerable number of years, germinated because of the strong effect that calculations and computations generated. Realistically, it is still quite common to see buildings with ornate knots or braid lattice-work. However, as a starting point for a study of the mathematics of knot, we need to excoriate this aesthetic layer and concentrate on the shape of the knot. Knot theory, in essence, is the study of the geometrical aspects of these shapes. Not only has knot theory developed and grown over the years in its own right, but also the actual mathematics of knot theory has been shown to have applications in various branches of the sciences, for example, physics, molecular biology, chemistry, et cetera.

In these lessons, we aim to guide the readers over the multifarious but non-technical aspects that make up the theory of knots. Throughout these lessons, we shall concentrate on lucid expositions, and most of the exercises that can be found within these lessons can be done 'physically', that is, by deforming a piece of closed string under some restrictions that I will explain accordingly in the lessons.

To enjoy the lessons, please make sure you have a piece of closed strings (without any open endings) in your hand... Enjoy!

Lesson 1    Introduction

Lesson 2    Fundamental Concepts of knot theory

Lesson 3    Fundamental Concepts of knot theory (continued)

Lesson 4    Knot invariants: Classical theory

Lesson 5    Knot invariants: Classical theory (continued) and Jones polynomial

Lesson 6    More fun stuff

Lesson 7    Applications

Reference:

[Li] An introduction to knot theory, (1997), Springer (Graduate text in mathematics 175), W. Lickorish

[Mu] Knot theory and its applications, (1996), Birkhauser, Kunio Murasugi

[BZ] Knots, (2002), De Gruyter studies in Mathematics G. Burde and H. Zieschang

[Kan] Examples on polynomial invariants of knots and links, (1986) Math. Ann. 275 pp.555-572 T. Kanenobu

[Sc] Die eindeutige Zerlegbarkeit eines Knotens in Primknoten, (1949) Springer H. Schubert

This work was made possible due to a grant from the National Science Foundation.