On Khovanov homology and related invariants

On Khovanov homology and related invariants

This paper begins with a survey of some applications of Khovanov homology to low-dimensional topology, with an eye toward extending these results to sl(n) homologies. We extend Levine-Zemke’s ribbon concordance obstruction from Khovanov homology to sl(n) foam homologies for n ≥ 2, including the universal sl(2) and sl(3) foam homology theories. Inspired by Alishahi and Dowlin’s bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and Bar-Natan spectral sequences also provide lower bounds on Turaev genus.

book chapter

Caprau, C., González, N., Lee, C. R. S., Lowrance, A. M., Sazdanović, R., & Zhang, M. (2021). On Khovanov homology and related invariants. In B. Acu, C. Cannizzo, D. McDuff, Z. Myer, Y. Pan, & L. Traynor (Eds.), Research Directions in Symplectic and Contact Geometry and Topology (Vol. 27, pp. 273–292). Springer International Publishing. https://doi.org/10.1007/978-3-030-80979-9_6