Topics for the exam

The exam consists on a 60 minutes long seminar on one of the topics below. Topics must be booked: any topic can be chosen by at most one person. Further topics can be proposed but they must be approved first - please write me an email if you wish to propose a topic. The talks marked with [M] are reserved for master students. (Master students are free to choose any topic not only those marked with [M]).

Fundations of Khovanov homology

• Functoriality of Khovanov homology [M]

Bar-Natan, Dror. “Khovanov’s Homology for Tangles and Cobordisms.” Geometry & Topology 9, no. 3 (2005): 1443–99. https://doi.org/10.2140/gt.2005.9.1443.

Various descriptions of Khovanov and Khovanov-Rozansky homologies

• Universal sl(3) homology via foams [PRENOTATO]

Mackaay, Marco and Vaz, Pedro.“The universal sl3–link homology” Algebraic & Geometric Topology 7, no. 3 (August 9, 2007): 1135–69. https://doi.org/10.2140/agt.2007.7.1135

• sl(3) homology via foams and via matrix factorisations

 Mackaay, Marco, and Pedro Vaz. “The foam and the matrix factorization sl3–link homologies are equivalent.” Algebraic & Geometric Topology 8, no. 1 (March 12, 2008): 309–42. https://doi.org/10.2140/agt.2008.8.309.

• An approach to sl(N) homology via foams

Mackaay, Marco, Marko Stošić, and Pedro Vaz. “sl(N)–Link Homology (N ≥ 4) Using Foams and the Kapustin–Li Formula.” Geometry & Topology 13, no. 2 (January 27, 2009): 1075–1128. https://doi.org/10.2140/gt.2009.13.1075.

• Comparison between gl(2) and sl(2) theories

Ehrig, Michael, Catharina Stroppel, and Daniel Tubbenhauer. “Generic gl(2)-Foams, Web and Arc Algebras.” arXiv, October 22, 2020. https://doi.org/10.48550/arXiv.1601.08010

Categorifications of the HOMPTFLY polynomial

• HOMPTFLY homology

Khovanov, Mikhail, and Lev Rozansky. “Matrix Factorizations and Link Homology II.” Geometry & Topology 12, no. 3 (June 4, 2008): 1387–1425. https://doi.org/10.2140/gt.2008.12.1387.

• Hochschild cohomology and triply graded theory

Khovanov, Mikhail. “Triply-Graded Link Homology and Hochschild Homology of Soergel Bimodules.” International Journal of Mathematics 18, no. 08 (September 2007): 869–85. https://doi.org/10.1142/S0129167X07004400

Applications of Khovanov and Khovanov-Rozansky homologies

• Other concordance invariants from Khovanov homology [M] 

Lewark, Lukas. “A New Concordance Homomorphism from Khovanov Homology.” arXiv, January 16, 2024. http://arxiv.org/abs/2401.08480

• The s-invariant for links in RP3 [PRENOTATO]

Manolescu, Ciprian, and Michael Willis. “A Rasmussen Invariant for Links in  RP3.” arXiv, November 19, 2024. https://doi.org/10.48550/arXiv.2301.09764.

• Distinguishing surfaces with Khovanov homology [M]

Sundberg, Isaac, and Jonah Swann. “Relative Khovanov–Jacobsson Classes.” Algebraic & Geometric Topology 22, no. 8 (March 14, 2023): 3983–4008. https://doi.org/10.2140/agt.2022.22.3983.

• The generalisation of Rasmussen's s-invariant to sl(n) [PRENOTATO]

Lobb, Andrew. “A Slice Genus Lower Bound from sl(n) Khovanov–Rozansky Homology.” Advances in Mathematics 222, no. 4 (November 2009): 1220–76. https://doi.org/10.1016/j.aim.2009.06.001.

• Other concordance invariants from Khovanov-Rozansky homologies I

Lewark, Lukas, and Andrew Lobb. “New Quantum Obstructions to Sliceness.” Proceedings of the London Mathematical Society 112, no 1 (January 2016): 81–114. https://doi.org/10.1112/plms/pdv068.

• Other concordance invariants from Khovanov-Rozansky homologies II

Lewark, Lukas, and Andrew Lobb. “Upsilon-like Concordance Invariants from sl(n) Knot Cohomology.” Geometry & Topology 23, no 2 (April 8, 2019): 745–80. https://doi.org/10.2140/gt.2019.23.745.

• Khovanov homology and strongly invertible links [PRENOTATO]

Sano, Taketo. “Involutive Khovanov Homology and Equivariant Knots.” arXiv, September 9, 2024. http://arxiv.org/abs/2404.08568.

• Khovanov homology and transverse knots

Plamenevskaya, Olga. “Transverse Knots and Khovanov Homology.” Mathematical Research Letters 13, no 4 (2006): 571–86. https://doi.org/10.4310/MRL.2006.v13.n4.a7.