Knot Homology

Summer Semester 2024

This is the website for the lecture course on Knot homology and its associated exercise class during the summer semester of 2024. The course was part of the Simons semester: Knots, Homologies and Physics.

This course was aimed at (advanced) Master students and PhD students specializing in algebra, category theory, geometry, topology, and adjacent areas of mathematical physics. Exceptionally motivated Bachelor students were also welcome. The purpose of the course was to bridge the gap between a first exposure to knot homology theories (excellent introductory articles and recorded mini-lecture series exists, see below) and the level of current research. The topic continued the theme of the ZMP Seminar on Knot homology from the winter semester 23/24, but having attended this was not a prerequisite.

Content

This course gives an extended introduction to knot homology theories and, more broadly, categorification in quantum topology.

The course was structured into the following parts:

Basic knot theory
The Jones polynomial
Cobordisms and TQFTs
Graded and homological algebra
A first look at Khovanov homology
Categorical background
Khovanov homology for tangles
Lee homology and the Rasmussen invariant
The colored Jones polynomials and their categorifications
Categorical projectors and the Rozansky-Willis invariant
Hecke algebras and HOMFLYPT
Quantum Schur-Weyl duality and gl(N) skein theory
Soergel bimodules
Rouquier complexes and Rickard complexes
Triply-graded link homology

Prerequisites

Prerequisites: it was assumed that participants had prior exposure to the following topics and concepts:

Algebra (incl. homological): groups, rings, modules, chain complexes, homotopy equivalence, homology, Ext, Tor
Topology (differential and algebraic): point-set topology, manifolds, orientations, fundamental group, homology, cohomology
Category theory: limits, colimits, monoidal structures, enriched categories

Coordinates

Lectures were held in person, streamed and recorded:

Wednesday, 16:15-17:45, Geomatikum H1, 3rd Apr 24 - 10th Jul 24
Friday, 12:15-13:45, Geomatikum H4, 5th Apr 24 - 12th Jul 24

Exercise classes were in person for participants in Hamburg:

Friday, 16:15-17:45, Room 434, Geomatikum, 12th Apr 24 - 12th Jul 24.

Ressources

Video recordings of lectures have been made accessible for registered participants and are available on request.

Introductory video resources:

Morrison, Scott. Khovanov homology (MSRI introductory workshop on link homology 2010) (Part 1, Part 2)
Rasmussen, Jacob. Introduction to Knot Theory (IAS/PSMI Lecture Series 2019) (Playlist, see esp. lectures 2-4)

Some background references (to be updated):

Bar-Natan, Dror. On Khovanov's categorification of the Jones polynomial. Algebr. Geom. Topol. 2 (2002), 337--370 (doi:10.2140/agt.2002.2.337)
Bar-Natan, Dror. Khovanov's homology for tangles and cobordisms. Geom. Topol. 9 (2005), 1443--1499 (doi:10.2140/gt.2005.9.1443)
Bar-Natan, Dror. Fast Khovanov homology computations. J. Knot Theory Ramifications 16 (2007), no. 3, 243--255 (doi:10.1142/S0218216507005294)
Khovanov, Mikhail. A categorification of the Jones polynomial. Duke Math. J. 101 (2000), no. 3, 359--426 (doi:10.1215/S0012-7094-00-10131-7)
Khovanov, Mikhail. Triply-graded link homology and Hochschild homology of Soergel bimodules. Internat. J. Math. 18 (2007), no. 8, 869--885 (doi:10.1142/S0129167X07004400)
Morrison, Scott; Walker, Kevin; Wedrich, Paul. Invariants of 4-manifolds from Khovanov-Rozansky link homology. Geom. Topol. 26 (2022), no. 8, 3367--3420 (doi:10.2140/gt.2022.26.3367)

Exercise / Exam

Exercise:

The preparatory meeting for the exercise classes took place on 5th Apr 24, 16:15, Room 434.

Exam:

For participants in Hamburg there will be an oral exam. To qualify for the exam, you should solve at least 50% of the homework problems and present on the board at least twice.