Heegaard Floer homology is an invariant of 3-manifolds, and knots and links within them, introduced by P. Oszváth and Z. Szabó in the early 2000s. Because of its relative computability by the standards of gauge and Floer theoretic invariants, it has enjoyed considerably popularity. However, it is not immediately obvious from the construction that Heegaard Floer homology is natural, that is, that it assigns to a basepointed 3-manifold a well-defined module over an appropriate base ring rather than an isomorphism class of modules, and well-defined cobordism maps to 4-manifolds with boundary. This situation was improved in the 2010s when A. Juhász, D. Thurston, and I. Zemke showed naturality of the various versions of Heegaard Floer homology. In this talk we consider involutive Heegaard Floer homology, a refinement of the theory introduced by C. Manolescu and I in 2015, whose definition relies on Juhász-Thurston-Zemke naturality but which is itself not obviously natural even given their results. We prove that involutive Heegaard Floer homology is a natural invariant of basepointed 3-manifolds together with a framing of the basepoint, and has well-defined maps associated to cobordisms, and discuss some consequences and implications. This is joint work with J. Hom, M. Stoffregen, and I. Zemke.

Hedden and Pinzón-Caicedo conjectured that every non-constant, winding number zero, satellite operator on the knot concordance group must admit an extreme form of rank-expansion, namely, they map a rank-one subgroup to an infinite rank subgroup. In this talk, we will give the first known example of such a phenomenon by showing that the Whitehead doubles admit such a behavior. In fact, we will show that the conjecture holds for a larger class of satellite knots defined using Conway tangles. We will also recover many existing results on rank-conservation in the literature proved using gauge theoretic methods. Our primary tool is a surgery formula in involutive Heegaard Floer homology proved by Hendricks-Hom-Stoffregen and Zemke. This is joint work with Irving Dai, Matt Hedden and Matt Stoffregen.

To a knot K in the 3-sphere we can associate the 3-manifold that arises from zero-framed Dehn surgery on K. It is a natural question to ask: if two knots have zero-surgeries which are Z-homology cobordant via cobordism W (with technical condition that the two positive knot meridians are homologous) does that imply that the knots must be concordant? In this talk, we give a pair of knots which are rationally concordant and whose zero-surgeries have an aforementioned homology cobordism between them, but the knots are not smoothly concordant. One knot in the pair is the figure eight knot, 4_1, which represents a 2-torsion element in the smooth concordance group C; all knots used in previous counterexamples represent infinite order elements. We will also discuss some related results involving iterating the pattern on 4_1 and other knots.

In this talk, I will describe two variations of Khovanov homology for null homologous links in RP^3 from two different perspectives: First, we will introduce a Bar-Natan deformation of the homology theory and define the corresponding s-invariant, which provides a genus bound for a certain class of slice surfaces. Second, we will explore an extension of the Ozsváth-Szabó spectral sequence on Heegaard Floer homology of branched double covers for null homologous links in RP^3, and generalize the E_2 page of it.

We review some conjectures on the Khovanov homology of torus knots and their relationship to modules over Khovanov’s arc algebra, and discuss Khovanov spectra. We then calculate the stable Khovanov spectra of three-stranded torus knots, and constrain Khovanov spectra of other torus knots. This is joint work with Mike Willis.

A question in knot theory that has recently become popular is to classify what knots bound a smooth disc in X - int(B^4), where X is a given closed 4-manifold. We study the case when X is the K3 surface, and prove that every knot with unknotting number less than 22 bounds a smooth disc in K3 - int(B^4). Our proof is constructive and based on the existence of a plumbing tree of 22 spheres in K3. This is joint work with Stefan Mihajlović.

Donald and Vafaee constructed a knot slicing obstruction for knots in the three-sphere by producing a bound relating the signature and second Betti number of a spin 4-manifold whose boundary is zero-surgery on the knot. Their bound relies on Furuta's 10/8 theorem and can be improved with the 10/8 + 4 theorem of Hopkins, Lin, Shi, and Xu. I will explain how to expand on this technique to obtain four-ball genus bounds and compute the bounds for some satellite knots. This is joint work in progress with Sashka Kjuchukova and Gordana Matic.

One of the key topics of study in smooth 4-manifold theory is to understand how submanifolds are embedded in 4-manifolds. In this talk we will discuss about exotic codim-1 embeddings, i.e. those which are topologically isotopic but not smoothly in 4-manifolds. I will try to propose a few motivating questions and explain some new ideas on how to find such interesting embeddings in 4-manifolds with small Betti numbers. This is a joint work with Hokuto Konno and Masaki Taniguchi.

Given a link in the thickened annulus, you can construct an associated link in a closed 3-manifold through a double branched cover construction. In this talk we will see that perspective on annular links can be applied to show annular Khovanov homology detects certain braid closures. Unfortunately, this perspective does not capture all information about annular links. We will see a shortcoming of this perspective inspired by the wrapping conjecture of Hoste-Przytycki. This is partially joint work with Fraser Binns.

Rasmussen's invariant for links in S^3 gives a way to extract topological information from the combinatorial constructions of Khovanov and Lee, leading to statements about surfaces in B^4. One would like to generalize this to links in other 3-manifolds, and get statements about surfaces in other 4-manifolds. I will give an overview of the theory and one approach to such a generalization, leading to similarly combinatorial Rasmussen invariants for links in S^1\times S^2 and \mathbb{RP}^3, and corresponding statements about surfaces in some related 4-manifolds.

We will discuss new extensions and interesting perspectives on the Heegaard Floer knot and link surgery formulas of Manolescu, Ozsvath and Szabo. Many of these perspectives are motivated by the theory of bordered Heegaard Floer theory due to Lipshitz, Ozsvath and Thurston. We will discuss applications of the theory, as well as future directions.

I will explain a construction of a pair of smooth, oriented 4-manifolds that are homotopy equivalent but not (stably) homeomorphic. This joint work with Daniel Kasprowski and Arunima Ray completes a programme outlined by Richard Stong in the 90s.