Abstract: Khovanov homology is a link invariant which categorifies Jones polynomial. It admits a skein sequence (which, roughly speaking, categorifies the skein relation defining Kauffman bracket): this skein sequence relates the homology of a diagram $D$ with the homology of $D_A$ and $D_B$, where $D_A$ (resp. $D_B$) represents the diagram obtained after performing an A (resp. B) smoothing to one of the crossings in $D$.

In this talk we will review the construction of the skein sequence for Khovanov homology following Viro's approach, and present some examples on how to wisely choose a crossing in $D$ in order to derive results on the Khovanov homology of several families of links.

Abstract: Odd Khovanov homology is a homological invariant of links that categorifies the Jones polynomial. It is the “evil twin” of Khovanov homology: while the two theories categorify the Jones polynomial and coincide over Z/2Z, they differ over Z. In some vague sense, odd Khovanov homology is a “super”, or anti-commutative, analogue of its even counterpart.

In this talk, we will survey what odd Khovanov homology is, and then explain a new construction of odd Khovanov homology (joint with Pedro Vaz). It gives an extension to tangles, but also makes precise what is so “odd” about odd Khovanov homology.

Abstract: Khovanov homology admits a deformation known as Lee homology which despite having uninteresting homology can actually be used to obtain interesting topological information via Rasmussen's s-invariant. Both theories can be formulated combinatorially via foam categories: to speak of Lee homology in this context, one needs the tool of idempotent completion for 1-categories.

In this talk, we move one categorical level up and explore these notions via 2-categories of deformed foams and 2-categorical idempotent completion.

Abstract: Knot Floer homology is a powerful invariant that captures subtle geometric information about knots. Recent work by Hanselman, Rasmussen, and Watson gives a visual way to understand this theory: instead of working with algebraic complexes, one can represent a knot by a collection of immersed curves in a punctured torus. In this talk, we use this geometric viewpoint to study (p,q)-cable knots and their knot Floer torsion. We will see how the immersed curves make these computations more intuitive, and present a simple application to the unknotting number.