Georgia Institute of Technology College of Sciences

When a topological object admits a group action, we expect that our invariants reflect this symmetry in their structure. This talk will explore how link symmetries are reflected in three generations of related invariants: the Jones polynomial; its categorification, Khovanov homology; and the youngest invariant in the family, the Khovanov stable homotopy type, introduced by Lipshitz and Sarkar. In joint work with Matthew Stoffregen, we use Lawson-Lipshitz-Sarkar's construction of the Lipshitz-Sarkar Khovanov homotopy type to produce localization theorems and Smith-type inequalities for the Khovanov homology of periodic links.

I will discuss how a graph theoretic construction used by Hirasawa and Murasugi can be used to show that the commutator subgroup of the knot group of a two-bridge knot is a union of an ascending chain of parafree groups. Using a theorem of Baumslag, this implies that the commutator subgroup of a two-bridge knot group is residually torsion-free nilpotent which has applications to the anti-symmetry of ribbon concordance and the bi-orderability of two-bridge knots. In 1973, E. J. Mayland gave a conference talk in which he announced this result. Notes on this talk can be found online. However, this result has never been published, and there is evidence, in later papers, that a proper proof might have eluded Mayland.

Building on previous work of Rozansky and Willis, we generalise Rasmussen’s s-invariant to connected sums of $S^1 \times S^2$. Such an invariant can be computed by approximating the Khovanov-Lee complex of a link in $\#^r S^1 \times S^2$ with that of appropriate links in $S^3$. We use the approximation result to compute the s-invariant of a family of links in $S^3$ which seems otherwise inaccessible, and use this computation to deduce an adjunction inequality for null-homologous surfaces in a (punctured) connected sum of $\bar{CP^2}$. This inequality has several consequences: first, the s-invariant of a knot in the three-sphere does not increase under the operation of adding a null-homologous full twist. Second, the s-invariant cannot be used to distinguish $S^4$ from homotopy 4-spheres obtained by Gluck twist on $S^4$. We also prove a connected sum formula for the s-invariant, improving a previous result of Beliakova and Wehrli. We define two s-invariants for links in $\#^r S^1 \times S^2$. One of them gives a lower bound to the slice genus in $\natural^r S^1 \times B^3$ and the other one to the slice genus in $\natural^r D^2 \times S^2$ . Lastly, we give a combinatorial proof of the slice Bennequin inequality in $\#^r S^1 \times S^2$.

‘Koszul duality’ is a phenomenon which algebraists are fond of, and has previously been studied in the context of '(bordered) Heegaard Floer homology' by Lipshitz, Ozsváth and Thurston. In this talk, I shall discuss an occurrence of Koszul duality which links older constructions in Heegaard Floer homology with the bordered Heegaard Floer homology of three-manifolds with torus boundary. I shan’t assume any existing knowledge of Koszul duality or any form of Heegaard Floer homology.