(Akram Alishahi)Unknotting number is one of the simplest, yet mysterious, knot invariants. For example, it is not known whether it is additive under connected sum or not. In this talk, we will construct lower bounds for the unknotting number using two homological knot invariants: knot Floer homology, and (variants of) Khovanov homology. Unlike most lower bounds for the unknotting number, these invariants are not lower bound for the slice genus and they only vanish for the unknot. Parallely, we will discuss connections between knot Floer homology and (variants of) Khovanov homology. One main conjecture relating knot Floer homology and Khovanov homology is that there is a spectral sequence from Khovanov homology to knot Floer homology. If time permits, we will sketch an algebraically defined knot invariant, for which there is a spectral sequence from Khovanov homology converging to it. The construction is inspired by counting holomorphic discs, so we expect it to recover the knot Floer homology. This talk is based on joint works with Eaman Eftekhary and Nathan Dowlin. (Artem Kotelskiy)
We will describe a geometric
interpretation of Khovanov homology as Lagrangian Floer homology of two
immersed curves in the 4-punctured 2-dimensional sphere. The main
ingredient is a construction which associates an immersed
curve to a 4-ended tangle. This curve is a geometric way to represent
Khovanov (or Bar-Natan) invariant for a tangle. We will show that for a
rational tangle the curve coincides with the representation variety of
the tangle complement.
The construction is inspired by a
result of [Hedden, Herald, Hogancamp, Kirk], which embeds 4-ended
reduced Khovanov arc algebra (or, equivalently, Bar-Natan dotted
cobordism algebra) into the Fukaya category of the 4-punctured
sphere. The main tool we will use is a category of peculiar modules,
introduced by Zibrowius, which is a model for the Fukaya category of a
2-sphere with 4 discs removed. This is joint work with Claudius
Zibrowius and Liam Watson.
