Math 191 Introduction to Knot Theory

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Course information

This  is an introductory course in Knot Theory. There are no formal prerequisites, but some familiarity with linear and abstract algebra, as well as an ability to visualize objects in three dimensions is useful. The course  is  assessable to advanced undergraduate students.

One can imagine a knot as a continuous loop (e.g., made of very thin elastic rubber) in the three-dimensional space. Given a knot, one can ask: is it really knotted? I.e., can it be deformed into a ring (the trivial "knot") without making any cuts? More generally, given two knots, one wants to know whether  one of them can be deformed into the other. In order to answer such questions, we must introduce and be able to compute (e.g., numerical and polynomial) knot invariants. In this class, we will study many different invariants of knots and will see how they allow to distinguish knots.

Knot theory has many relations to topology, physics, and (more recently!) even the study of the structure of DNA. Some of these  connections  will be explored in the second part of the class.

Textbook:

"Knot knotes" by J.Roberts. The textbook is available at the University bookstore. Please, follow the above link for an electronic preview.

Recommended books:

There are a lot of different  books on Knot Theory. I list below several books which are perhaps the closest to the topics we will study in class and are available at the UCLA library.

1. C. Adams "The knot book".
2. L. Kauffmann "Knots and Physics".
3. D. Rolfsen "Knots and Links".
4. V.V. Prasolov, A.B. Sossinsky "Knots, Links, Braids and 3-manifolds" .
5. C. Livingston "Knot theory".

Homework Assignments

The class will have weekly homework which will be announced in class and send to you by e-mail. The homework is due on Friday. Please, note that two lowest homework scores will be dropped from the computation of your grade.

Midterm

We will have one midterm examination on the date February 15th (Please, note that the date is subject to change).