Special topics course offered in Fall 2018 by Caitlin Leverson on "Introduction to Knot Theory"
MATH 1553, MATH 1554, MATH 1564 or permission from the instructor.
The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots by Colin C. Adams.
Class notes distributed in class.
This course will explore the beautiful mathematics of knots which are pieces of string that are knotted in the colloquial sense and then have their ends glued together. Given two such knots, it is surprisingly interesting and difficult to decide whether or not you can rearrange one knot into the other (no scissors allowed). This course will look at different ways to represent these knots and various techniques for deciding whether one knot can be rearranged into another. We will also explore the applications of knot theory to biology, chemistry, and physics.
1. Knot theory basics
(a) Definition of knots and links
(b) Composition of knots
(c) Reidemeister moves
2. Things which lead to knots
3. Knot tabulation
(a) Dowker notation
(b) Conway notation and tangles
(c) Knots and planar graphs
4. Knot invariants
(a) Unknotting number
(b) Bridge number
(c) Crossing number
(d) Genus and Seifert surfaces
5. Polynomial invariants of knots
(a) Bracket and Jones polynomials
(b) Alexander and HOMFLY-PT polynomials
(a) Biology and DNA
(b) Chemistry and knotted molecules
(c) Physics and statistical mechanics