Math 690: Knot Homologies

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 certainĀ 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.

Lecture videos:

  1. January 21: basic definitions, fundamental group, Alexander polynomial
  2. January 26: Seifert forms, Alexander-Conway polynomial, slice knots
  3. January 28: Slice knots, concordance, Jones polynomial
  4. February 2: Jones polynomial, Tait conjectures. (I forgot to stop screen sharing, so the second half is hard to watch - apologies!)
  5. February 4: Khovanov homology: introduction
  6. February 9: Khovanov homology: Reidemeister invariance, basic properties
  7. February 11: Khovanov homology: cobordism maps, s invariant
  8. February 16: Applications of the s invariant, homotopy-slice and homology-slice knots, overview of knot Floer homology
  9. February 18: Knot Floer homology, tau invariant, Khovanov homology for knots in \( \#^r S^1 \times S^2 \)

Final Exam: Just kidding!