MAT 418 Introduction to Knot Theory
A knot is a continuous loop in the three-dimensional space. Given two knots, we are primarily interested in determining whether they are same or different, i.e., whether one can be deformed into the other. In this class, we will introduce different knot invariants and see how these various invariants of knots can be used to distinguish them. Inspiration for the study of knots comes from the physical knots that are present in our world, and applications arise in biology, computer science and physics. Topics that we will discuss include composition of knots, Reidemeister moves, links, tricolorability, knots and planar graphs, unknotting number, crossing number, genus and Seifert surfaces, braids, bracket polynomial and Jones polynomial, the Alexander and HOMFLY polynomials, applications to biology, knots in graphs, prime decomposition of knots.