This course will cover the basics of knot theory. A (mathematical) knot can be thought of as a piece of string which has been knotted (in the traditional sense) with its ends glued. Two knots are the "same" if one can be moved through space to look exactly like the other (without breaking the gluing). An essential question in knot theory is to be able to distinguish different knots. Answering this question proves challenging and the attempt to address it has inspired beautiful mathematics for hundreds of years, and continues today. To distinguish knots, mathematicians use "invariants." We will discuss various ways to present a knot, invariants which can be used to distinguish them, and applications of knot theory to low-dimensional topology more broadly and to the sciences.
The course will cover the following main themes:
1. Knot theory basics: definitions of knots/links and isotopy
2. Presentations of knots: diagrams, tangles, braids
3. Knot invariants: colorability, polynomials, unknotting number, Seifert genus
4. Knots and computers: dowker notation, algorithms in knot theory
5. Applications: knots/links in low-dimensional topology, knots in the sciences