Knot homology theories

Math 603D

Fall 2020

This is a course on homological invariants of knots. This will take Khovanov homology as a central object of study, with a focus on the current state of homological invariants in low-dimensional topology, more generally, since Khovanov's initial work categorifying the Jones polynomial [Kh2000], which was posted to the arXiv just over 20 years ago.

As a starting point, this course will assume a little basic knot theory, in particular, the definition of a knot. A great reference for this is Dale Rolfsen's classic book Knots and Links [Ro1976].



[BN2002] Dror Bar-Natan On Khovanov's categorification of the Jones polynomial Algebr. Geom. Topol. (2002)

[BN2005] Dror Bar-Natan Khovanov's homology for tangles and cobordisms Geom. Topol. (2005)

[BN2007] Dror Bar-Natan Fast Khovanov homology computations J. Knot Theory Ramifications (2007)

[Bl2010] Jonathan Bloom Odd Khovanov homology is mutation invariant Math. Res. Lett. (2010)

[HKK2017] F Haiden, L Katzarkov, and M Kontsevich Flat surfaces and stability structures Publ. Math. Inst. Hautes Études Sci. (2017)

[HRW2017] Jonathan Hanselman, Jacob Rasmussen, and Liam Watson Bordered Floer homology for manifolds with torus boundary via immersed curves arXiv.1604.03466 (2016)

[Jo1987] Vaughan Jones Hecke algebra representations of braid groups and link polynomials Ann. of Math. (1987)

[Kh2000] Mikhail Khovanov A categorification of the Jones polynomial Duke Math. J. (2000)

[Kh2006] Mikhail Khovanov Link homology and Frobenius extensions Fund. Math. (2006)

[KWZ2019] Artem Kotelskiy, Liam Watson, and Claudius Zibrowius Immersed curves in Khovanov homology arXiv:1910.14584 (2019)

[Lee2005] Eun Soo Lee An endomorphism of the Khovanov invariant Adv. Math. (2005)

[MOz2008] Ciprian Manolescu and Peter Ozsváth On the Khovanov and knot Floer homologies of quasi-alternating links. Gökova Geometry/Topology Conference (2008)

[Pi2020] Lisa Piccirillo The Conway knot is not slice Ann. of Math. (2020)

[Ra2005] Jacob Rasmussen Knot polynomials and knot homologies in Geometry and topology of manifolds, Fields Inst. Commun. (2005)

[Ra2010] Jacob Rasmussen Khovanov homology and the slice genus Invent. Math. (2010)

[Ro1976] Dale Rolfsen Knots and Links Publish or Perish Press (1976)

[We2010] Stephan Wehrli Mutation invariance of Khovanov homology over F2 Quantum Topol (2010)