TOPICS IN KNOT THEORY

Course Description

This course introduces the mathematical study of knots and links focusing on how topology, algebra, and geometry interact to distinguish and classify them. Students will learn both classical and modern knot invariants. A more apt title for this course could be "Topics in Geometric Topology," as we will also touch upon the importance of knots and links in the broader field of low-dimensional topology. Possible topics include the study of codimension one embeddings in manifolds, the knot group, Seifert surfaces and Seifert genus, polynomial link invariants, braids, finite and infinite cyclic coverings, the Alexander module, surgery theory, hyperbolic knot theory, and quantum invariants. The course is suitable for students with a background in linear algebra, abstract algebra, and point-set topology.

Required Text

Przytycki, Bakshi, Ibarra, Montoya-Vega, and Weeks, Lectures in Knot Theory: An Exploration of Contemporary Topics Springer Universitext

Optional Text

Rolfsen, Dale, Knots and Links AMS Chelsea Publishing