MAT 402 - Knot Theory (Spring 2020)

Textbook

Richard H. Crowell. Ralph H. Fox, Introduction to Knot Theory, Springer-Verlag (1963)

Course Description

This course is an introduction to Knot Theory at the undergraduate level. A mathematical knot is an

embedded circle in Euclidean space. The basic problem of knot theory is to determine when two knots

are equivalent, and this leads to the study of several different kinds of “knot invariants” - i.e., mathematical

objects that can be used to distinguish knots.

Although knots may appear innocuous at first, their theory is very rich and becomes increasingly complicated

as one delves deeper. Knot theory is a subarea of Topology, and as such it is impossible to probe

without a solid understanding of concepts such as topological spaces, continuity, compactness and connectedness.

Moreover, knot invariants are usually algebraic in nature, so a good background in algebra

(group theory especially) is also important.

We will not deal with knots directly in the first part of the course; rather, we will spend the first weeks

building up the necessary background in algebraic topology and group theory. In the second part we will

apply what we learned to understand how certain knot invariants - the knot group and knot polynomials

- can be used to distinguish between knots.

This seminar course is aimed primarily at third- and fourth-year students majoring in mathematics. This

course satisfies the University SPK requirement - see the syllabus for more information. As such, after a couple

weeks of lectures I will assign topics to students to present in class. These topics will usually be taken

from the textbook, but students are also welcome to suggest topics they would be interested in presenting,

as long as it is related to knot theory. Each student will have to give two 30 minutes presentations during

the semester.