Reading course on Khovanov homology

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).

Program


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

Course materials

  1. [Kho] M. Khovanov. A categorification of the Jones polynomial. Duke Math. J. 101 (2000), no. 3, 359--426 arXiv version
  2. [BN] D. Bar-Natan. On Khovanov's categorification of the Jones polynomial. Algebraic and Geometric Topology 2 (2002) 337-370. arXiv version
  3. [Lee] E. S. Lee. An endomorphism of the Khovanov invariant. Adv. Math. 197 (2005), no. 2, 554-586. arXiv version
  4. [Ras] J. Rasmussen. Khovanov homology and the slice genus. Invent. Math. 182 (2010), no. 2, 419-447. arXiv version
  5. [Pic] L. Piccirillo. The Conway knot is not slice. 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. M. Jacobsson. An invariant of link cobordisms from Khovanov homology. Algebr. Geom. Topol. 4 (2004) 1211-1251. arXiv version
  5. D. Bar-Natan. Khovanov's homology for tangles and cobordisms. Geom. Topol. 9(2005) 1443-1499. arXiv version
  6. Topics course on Khovanov homology (Spring 2018): MAT 280