MAT 280, Spring 2018


Instructor: Eugene Gorsky, egorskiy AT
Office hours: TR 2-3pm at Math 2113
If you have a question and cannot come at office hours, write me an email to schedule an appointment.

General course information


  1. Skein relations and knot polynomials. Alexander polynomial, Jones polynomial, Kauffman's bracket. State-sum models.
  2. Khovanov homology: construction, proof of invariance. Examples, relation to the Jones polynomial.
    Skein exact sequence. Properties of Khovanov homology
  3. Khovanov-Lee homology: construction and Lee's spectral sequence.
  4. Khovanov-Lee homology: computation, Rasmussen's s-invariant. Properties of the s-invariant.
  5. Cobordisms, 4-ball genus and Rasmussen's proof of the Milnor conjecture.
If time permits, we may cover more recent developments in Khovanov homology.

Course materials

There will be no textbook for the course, it will be mainly based on the following papers:
  1. M. Khovanov. A categorification of the Jones polynomial. Duke Math. J. 101 (2000), no. 3, 359--426 arXiv version
  2. D. Bar-Natan. On Khovanov's categorification of the Jones polynomial. Algebraic and Geometric Topology 2 (2002) 337-370. arXiv version
  3. E. S. Lee. An endomorphism of the Khovanov invariant. Adv. Math. 197 (2005), no. 2, 554-586. arXiv version
  4. J. Rasmussen. Khovanov homology and the slice genus. Invent. Math. 182 (2010), no. 2, 419-447. arXiv version
  5. M. Jacobsson. An invariant of link cobordisms from Khovanov homology. Algebr. Geom. Topol. 4 (2004) 1211-1251. arXiv version
Further references:
  1. O. Viro. Remarks on definition of Khovanov homology. Fund. Math. 184 (2004), 317-342. arXiv version
  2. M. Asaeda, M. Khovanov. Notes on link homology. arXiv version
  3. P. Turner. Five Lectures on Khovanov Homology. arXiv version
  4. D. Bar-Natan. Khovanov's homology for tangles and cobordisms. Geom. Topol. 9(2005) 1443-1499. arXiv version

Grading policy

The students are expected to choose (after discussing with the instructor) a research paper about Jones polynomial, Khovanov homology or related topics and give a short presentation in class about it. Alternatively, they can submit a short written report about the paper.

Some possible papers for presentations:

Disability Services

Any student with a documented disability who needs to arrange reasonable accommodations must contact the Student Disability Center (SDC). Faculty are authorized to provide only the accommodations requested by the SDC. If you have any questions, please contact the SDC at (530)752-3184 or