Recent techniques in Floer and Khovanov homology

Abstract: In a recent paper, Juhasz, Miller and Zemke proved an inequality involving the number of local maxima and the genus appearing in an oriented knot cobordism using a version of knot Floer homology. In this talk I will be discussing some similar inequalities for non-orientable knot cobordisms using the torsion orders of unoriented versions of knot Floer homology and instanton Floer homology. This is a joint work with Marco Marengon.

The monopole Floer homology of a closed oriented 3-manifold was introduced by Kronheimer-Mrowka around 2007 and has greatly influenced the study of 3-manifold topology since its inception. In this talk, we will generalize their construction and define the monopole Floer homology for any pair (Y, w), where Y is any compact oriented 3-manifold with toroidal boundary and w is a suitable closed 2-form. The graded Euler characteristic of this Floer homology recovers the Milnor-Turaev torsion invariant by a classical theorem of Meng-Taubes. It satisfies a reasonable (3+1) TQFT property. In the end, we will explain its relation with gauged Landau-Ginzburg models and point out some future directions.

Abstract: In this talk, I'll provide an upper bound and a lower bound on the dimension of SHI. In the first part, I'll review the constructions of sutured (Heegaard) Floer homology SFH by Juhász, sutured monopole homology SHM and sutured instanton homology SHI by Kronheimer-Mrowka. In the second part, I'll show dim KHI is less or equal to dim SFC, where SFC is the chain complex of SFH for some Heegaard diagram. Also, I'll show graded Euler characteristics of SHI and SFH are the same, which provides a lower bound of dim SHI. As an application, I show KHI=HFK^hat for (1,1)-L-space knots (in particular torus knots) and constrained knots (a generalization of 2-bridge knots in lens spaces). This is joint work with John A. Baldwin and Zhenkun Li.

Abstract: In this talk, I will present an immersed-curve approach to computing the knot Floer homology of a class of satellite knots. This class includes Whitehead doubles, cable knots, and Mazur satellite knots as special cases. 

Abstract: In part 1, we discuss immersed curves both as a method to compute Heegaard Floer homology and as a technique to Dehn surgery problems in low-dimensional topology. In part 2, we use a Heegaard Floer theoretic approach via immersed curves to study the Cabling Conjecture and thin knots. We show that thin, slice knots satisfy the conjecture using techniques for the general thin setting.

Abstract: This talk describes how immersed curves can be used to classify chain complexes and give the best known partial classification of the homology cobordism group.

Abstract: When restricted to alternating links, both Heegaard Floer and Khovanov homology are concentrated in a single δ-grading. This leads to the broader class of thin links. In a recent preprint, my collaborators Artem Kotelskiy and Liam Watson and I provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain multicurve invariants for Conway tangles that we developed in previous works.

Abstract: I will discuss some open problems, and survey some classical material regarding symmetric knots, and group actions on 3- and 4-manifolds. In the second part of the talk I will discuss how techniques from Floer theory can be employed to approach some of these problems. 

Abstract: A knot is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all infinite order. All previously known examples of rationally slice knots were order two. This is a joint work with Jennifer Hom, JungHwan Park, and Matthew Stoffregen.

Abstract: Since its introduction in 1966 by Fox and Milnor the knot concordance group has been an invaluable algebraic tool for examining the relationships between 3- and 4- dimensional spaces. Though knots generalize naturally to links, this group does not generalize in a natural way to a link concordance group. In talk on joint work with Matt Hedden, we define a link concordance group based on the ``knotification” construction of Ozsvath and Szabo. 

Abstract: In this talk, we will use the invariants from the Heegaard Floer theory to study possible links of singularities of algebraic curves in the complex projective plane. In particular, we will concentrate on curves with positive/ negative double points. 

Using Heegaard Floer homology, one can associate to a rational homology 3-sphere Y that is equipped with a spinc structure s, a rational number commonly referred to as the d-invariant of (Y, s). d-invariants have been useful in answering a range of questions in low-dimensional topology. A nice source of rational homology 3-spheres comes from considering branched double covers Sigma_2(K) of knots K in S^3. If Sigma_2(K) is an L-space, then the d-invariant of Sigma_2(K), at the unique spin-structure s, is well-understood: Lin-Ruberman-Saveliev in 2020 showed that it's a multiple of the signature of K.

When the branch set is a quasi-alternating link, the d-invariants of the branched double cover can be recovered from the signatures of the link in a similar way; this is due to Lisca-Owens in 2015. In this talk, we show that a similar phenomenon holds for branching over Montesinos links, many of which are not quasi-alternating. This is work in progress with M. Marengon.

Abstract: This two-part talk is based on my work with Carmen Caprau, Nicolle Gonzalez, Christine Ruey Shan Lee, Adam Lowrance, and Radmila Sazdanovic. Part 1 shows that sl(N) web/foam homologies give obstructions to ribbon concordance, and is inspired by work of Levine and Zemke. Part 2 uses spectral sequences to bound distance measurements between knots (alternation number and Turaev genus) and is inspired by work of Alishahi and Dowlin. 

Abstract: Two knots are homology concordant if they are smoothly concordant in a homology cobordism. The group CZ_hat (resp. CZ) was previously defined as the set of knots in homology spheres that bound homology balls (resp. in S3), modulo homology concordance. We prove the quotient CZ_hat/CZ contains a infinite rank free subgroup. We construct our family of examples by applying the filtered mapping cone formula to L-space knots, and prove linear independence with the help of the connected knot complex.

Abstract: The question of when there exists a rational homology cobordism between two lens spaces was completely answered by Lisca. Lisca’s proof relies on a combinatorial analysis of embeddings of certain intersection lattices into the standard Euclidean lattice. A refinement of the above question is to ask when there exists a ribbon rational homology cobordism from one lens space to another, i.e. one that can be built using just 1- and 2-handles. In these talks, I will show how the absence of 3-handles in a rational homology cobordism translates into a condition on the embeddings of intersection lattices, and, moreover, I will illustrate how this obstruction together with Lisca’s machinery can be used to completely determine when there exists a ribbon rational homology cobordism from a lens space to another.

Abstract: Kawauchi defined a group structure on the set of homology S1×S2’s under an equivalence relation called H~-cobordism. This group receives a homomorphism from the knot concor- dance group, given by the operation of zero-surgery. We apply knot concordance invariants derived from knot Floer homology to study the kernel of the zero-surgery homomorphism. As a consequence, we show that the kernel contains a Z∞-subgroup generated by topologically slice knots in the smooth category. Moreover, this group can be defined in the topological category. There is a surjective homomorphism from the group defined in the smooth category to that defined in the topological category. We prove that if a homology S1×S2 has the trivial Alexander polynomial, then it is contained in the kernel of the homomorphism by using Freedman and Quinn’s result about Z-homology 3-spheres.

Abstract: Given a closed 4-manifold X with an indefinite intersection form, we consider smoothly embedded surfaces in X-int(B^4), with boundary a given knot K in the 3-sphere. We give several methods to bound the genus of such surfaces in a fixed homology class. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disk) in a 4-manifold and show that the set of H-slice knots can detect exotic smooth structures on closed 4-manifolds.

This is joint work with Ciprian Manolescu and Lisa Piccirillo.