This is an informal reading course on Khovanov homology and its applications in 4-dimensional topology, running in Winter/Spring 2020. The key objective is to understand
Lisa Piccirillo's proof that Conway knot is not slice.

The course runs on Tuesdays 1-2pm in Math 3232 (Winter) and on Thursdays 2-3pm on Zoom (Spring).

1/14 (Tonie) Kauffman bracket, Jones polynomial and cube of resolutions. [BN] section 2, Adams Knot book

1/21 no class

1/28 Frobenius algebras and TQFT. [Kho] Sections 2.2-2.3, more pictures and details . Notes by Tonie

2/4 Khovanov complex: definition, gradings, examples. HW: compute Khovanov homology of the trefoil

[BN] Sections 3.1-3.2. Notes by Tonie

2/11 no class

2/18 Digression on homological algebra I: complexes, homology, Euler characteristic

Notes by Tonie

2/25 (Neetal) Lee homology: definition, base change, generators Notes by Tonie

3/3 Khovanov homology: proof of invariance. [BN] Section 3.5

Notes by Tonie

3/10 no class

Spring break

3/24 (Addie) Digression on homological algebra II: spectral sequences Notes by Addie

3/31 More on spectral sequences, Lee spectral sequence. Notes

4/7 Definition and basic properties of s-invariant Notes

4/16 Cobordisms and movie moves. Notes

4/23 (James) s-invariant and 4-genus bound. Rasmussen's proof of the Milnor conjecture. Notes , pictures

4/30 (James) More on cobordisms maps, examples of slice genus computations. Properties of s-invariant.

Notes part 1 , part 2 , part 3 .

5/7 (Laura) 4-dimensional Kirby calculus notes

5/14 (Lisa) Conway knot is not slice notes

5/21 Alexander polynomial and signature notes

5/28 (Laura) The trace embedding lemma and spinelessness. notes

- [Kho] M. Khovanov. A categorification of the Jones polynomial. Duke Math. J. 101 (2000), no. 3, 359--426 arXiv version
- [BN] D. Bar-Natan. On Khovanov's categorification of the Jones polynomial. Algebraic and Geometric Topology 2 (2002) 337-370. arXiv version
- [Lee] E. S. Lee. An endomorphism of the Khovanov invariant. Adv. Math. 197 (2005), no. 2, 554-586. arXiv version
- [Ras] J. Rasmussen. Khovanov homology and the slice genus. Invent. Math. 182 (2010), no. 2, 419-447. arXiv version
- [Pic] L. Piccirillo. The Conway knot is not slice. arXiv version

- 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
- M. Jacobsson. An invariant of link cobordisms from Khovanov homology. Algebr. Geom. Topol. 4 (2004) 1211-1251. arXiv version
- D. Bar-Natan. Khovanov's homology for tangles and cobordisms. Geom. Topol. 9(2005) 1443-1499. arXiv version
- Topics course on Khovanov homology (Spring 2018): MAT 280