Overview: This course will be a brief introduction to homological invariants of knots and links, with an emphasis on the powerful invariant known as Khovanov homology and its applications to 3- and 4-dimensional topology. We'll start with an overview of some classical knot theory, particularly the Alexander and Jones polynomials and the notion of concordance. We'll then launch into the construction and properties of Khovanov homology. Toward the end, we will discuss some noteworthy recent results, such as Piccirillo's proof that the Conway knot is not slice and Manolescu-Marengon-Sarkar-Willis's work on surfaces in certain 4-manifolds other than ordinary 4-space.
Prerequisites: Elementary algebraic topology, at the level of Duke's Math 611. (In fact, you mostly just need to know the basic ideas of homological algebra: chain complexes, homology, chain maps, etc.)
Times: Tuesdays and Thursdays, 1:45-3:00pm, January 21 - February 18.
Zoom link: https://duke.zoom.us/j/99273132013. Please contact me for the password. Links to recordings of the lectures will be posted on this page.
References: The main references for the course will be Dror Bar-Natan's two classic papers, On Khovanov's categorification of the Jones polynomial and Khovanov's homology for tangles and cobordisms. Some helpful background material (which will be discussed during the first two lectures) can be found in textbooks such as Dale Rolfen's Knots and Links and W.B. Raymond Lickorish's An Introduction to Knot Theory.
Final Exam: Just kidding!