**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 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**

**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.*