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
minilecture 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 RozanskyWillis invariant
 Hecke algebras and HOMFLYPT
 Quantum SchurWeyl duality and gl(N) skein theory
 Soergel bimodules
 Rouquier complexes and Rickard complexes
 Triplygraded link homology
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): pointset topology, manifolds, orientations, fundamental group, homology, cohomology
 Category theory: limits, colimits, monoidal structures, enriched categories
Resources:
Video recordings of lectures have been made accessible for registered participants and are available on request.
Introductory video resources:
Some background references (to be updated):
 BarNatan, Dror. On Khovanov's categorification of the Jones polynomial. Algebr. Geom. Topol. 2 (2002), 337370 (doi:10.2140/agt.2002.2.337)
 BarNatan, Dror. Khovanov's homology for tangles and cobordisms. Geom. Topol. 9 (2005), 14431499 (doi:10.2140/gt.2005.9.1443)
 BarNatan, Dror. Fast Khovanov homology computations. J. Knot Theory Ramifications 16 (2007), no. 3, 243255 (doi:10.1142/S0218216507005294)
 Khovanov, Mikhail. A categorification of the Jones polynomial. Duke
Math. J. 101 (2000), no. 3, 359426 (doi:10.1215/S0012709400101317)
 Khovanov, Mikhail. Triplygraded link homology and Hochschild homology
of Soergel bimodules. Internat. J. Math. 18 (2007), no. 8, 869885 (doi:10.1142/S0129167X07004400)
 Morrison, Scott; Walker, Kevin; Wedrich, Paul. Invariants of 4manifolds from KhovanovRozansky link homology. Geom. Topol. 26 (2022), no. 8, 33673420 (doi:10.2140/gt.2022.26.3367)
Coordinates:
Lectures were held in person, streamed and recorded:
 Wednesday, 16:1517:45, Geomatikum H1, 3rd Apr 24  10th Jul 24
 Friday, 12:1513:45, Geomatikum H4, 5th Apr 24  12th Jul 24
Exercise classes were in person for participants in Hamburg:
 Friday, 16:1517:45, Room 434, Geomatikum, 12th Apr 24  12th Jul 24.
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.
Contact:
Paul Wedrich.
