MAT 280, Spring 2018

Instructor

Instructor: Eugene Gorsky, egorskiy AT math.ucdavis.edu.
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

Program

Skein relations and knot polynomials. Alexander polynomial, Jones polynomial, Kauffman's bracket. State-sum models.
Khovanov homology: construction, proof of invariance. Examples, relation to the Jones polynomial.
Skein exact sequence. Properties of Khovanov homology
Khovanov-Lee homology: construction and Lee's spectral sequence.
Khovanov-Lee homology: computation, Rasmussen's s-invariant. Properties of the s-invariant.
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:

M. Khovanov. A categorification of the Jones polynomial. Duke Math. J. 101 (2000), no. 3, 359--426 arXiv version
D. Bar-Natan. On Khovanov's categorification of the Jones polynomial. Algebraic and Geometric Topology 2 (2002) 337-370. arXiv version
E. S. Lee. An endomorphism of the Khovanov invariant. Adv. Math. 197 (2005), no. 2, 554-586. arXiv version
J. Rasmussen. Khovanov homology and the slice genus. Invent. Math. 182 (2010), no. 2, 419-447. arXiv version
M. Jacobsson. An invariant of link cobordisms from Khovanov homology. Algebr. Geom. Topol. 4 (2004) 1211-1251. arXiv version

Further references:

O. Viro. Remarks on definition of Khovanov homology. Fund. Math. 184 (2004), 317-342. arXiv version
M. Asaeda, M. Khovanov. Notes on link homology. arXiv version
P. Turner. Five Lectures on Khovanov Homology. arXiv version
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:

J. Fassler. Braids, the Artin Group, and the Jones Polynomial. [Yue Zhao]
E. Witten. Quantum field theory and Jones polynomial. [ Niklas Garner ]
O. Plamenevskaya. Transverse knots and Khovanov homology. [ Beibei Liu ]
P. Turner. A spectral sequence for Khovanov homology with an application to (3,q)-torus links.
J. Batson, C. Seed. A Link Splitting Spectral Sequence in Khovanov Homology. [ Oscar Kivinen ]
Jones, V. F. R. Hecke algebra representations of braid groups and link polynomials. Ann. of Math. (2) 126 (1987), no. 2, 335-388. [ Jianping Pan ]
Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W. B. R.; Millett, K.; Ocneanu, A. A new polynomial invariant of knots and links.
A. Champanerkar, I. Kofman. Spanning trees and Khovanov homology. [Matthew Lin]
P. Ozsvath, J. Rasmussen, Z. Szabo. Odd Khovanov homology. [ Subhadip Dey ]
J. Grigsby, A. Licata, S. Wehrli. Annular Khovanov homology and knotted Schur-Weyl representations. [ Shuang Ming ]

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 sdc@ucdavis.edu.