Times below are in US Eastern Daylight Time (UTC-4).

 

Title and Abstracts

1- Samantha Allen: Relationships between untwisting number and similar invariants

  • Abstract: The untwisting number of a knot K is the minimum number of null-homologous full twists required to unknot K. We relate this invariant to many other invariants, including the rational unknotting number, the (algebraic) 2k-unknotting number (related to the 4-move conjecture(s)), and the (algebraic) surgery description number.  Some of these will give a refinement of an inequality (by McCoy) related to the topological 4-genus, which itself is a analogue of the relationship between Rasmussen’s s-invariant, smooth 4-genus, and unknotting number.  In this talk, I will provide the first known example where untwisting number and surgery description number are not equal and discuss challenges to distinguishing these invariants in general.  This work is joint with Kenan Ince, Seungwon Kim, Benjamin Ruppik, and Hannah Turner.

2- Marco De Renzi:   Quantum and homological representations of mapping class groups of surfaces

  • Abstract: Quantum topology provides highly organized families of invariants, whose definition is flexible and general. For some of them, a more classical homological reformulation is known. This often allows for an efficient control over the topological content of the resulting constructions, as witnessed by Bigelow’s spectacular proof of the linearity of braid groups. For the mapping class group Mod(Σ) of a surface Σ, we will explain how to recover the family of quantum representations associated with the small quantum group of sl(2) by a classical construction, with Mod(Σ) acting on the homology with twisted coefficients of configuration spaces of Σ. This is a joint work with Jules Martel.

3- Peter Feller:  On the values taken by slice torus invariants

  • Abstract:We consider the space of so-called slice torus invariants, knot invariants that include the celebrated Rasmussen s invariant and Ozsváth-Szabó s tau invariant. In particular we characterize the set of values that slice-torus invariants may take on a given knot in topological terms.A part of our study is motivated by the concept of squeezed knots, which is discussed in A. Lobb's talk. Based on joint work with L. Lewark and A. Lobb.

3- Eugene Gorsky: Triply graded link homology: results and structures

  • Abstract: Khovanov and Rozansky defined in 2005 a triply graded link homology theory which categorifies HOMFLY-PT polynomial. The definition is somewhat technical and intimidating, nevertheless, there was a lot of recent progress in our understanding of Khovanov-Rozansky homology. In this talk, I will review some concrete examples of Khovanov-Rozansky homology computed for various families of links, as well as the general structures such as symmetry and homological operations. The talk is bases on joint works with Alex Chandler (in progress), Matt Hogancamp and Anton Mellit.

4- Matthew Hogancamp: A Kirby color for Khovanov homology

  • Abstract: In this talk I will discuss how to construct an object of the annular Bar-Natan category (rather, a completion thereof) which satisfies handle-slide invariance. This work is joint with Dave Rose and Paul Wedrich, and is intended to yield a chain-level refinement of the Manolescu-Neithalath 2-handle formula for skein lasagna modules.

5- Mikhail Khovanov: Facets of the universal construction

  • Abstract: Universal construction of topological theories starts with an evaluation of closed n-manifolds or, more generally, n- dimensional objects, and build a functor from a suitable category of n-cobordisms to the category of modules over a commutative ring. We'll review how universal construction for foams naturally emerges from link homology and then look at recent explorations of the universal construction in dimensions 1 and 2, including for manifolds with defects, leading to: 

         (a) a deformation of the Deligne category of generic representations of the symmetric group (joint work with R.Sazdanovic)   and study of these deformations (joint with V.Ostrik and Y.Kononov) 

         (b) an approach to biadjoint functors via evaluations of planar diagrams of nested circles (joint with R.Laugwitz). 

         (c) construction of rigid categories from regular languages and relation to automata (joint with M.S.Im). 

 

6- Adam Levine: Khovanov homology and cobordisms between split links

  • Abstract: We discuss the (in)sensitivity of the Khovanov functor to four-dimensional linking of surfaces. We prove that if L and L' are split links, and C is a cobordism between L and L' that is the union of disjoint (but possibly linked) cobordisms between the components of L and the components of L', then the map on Khovanov homology induced by C is completely determined by the maps induced by the individual components of C and does not detect the linking between the components. As a corollary, we prove that a strongly homotopy-ribbon concordance (i.e., a concordance whose complement can be built with only 1- and 2-handles) induces an injection on Khovanov homology. This is joint work with Onkar Singh Gujral.

6- Lukas Lewark: Khovanov homology and rational unknotting

  • Abstract: We will see a new geometric application of Khovanov homology. Specifically, we'll work with a certain universal variation of Khovanov homology, which associates to a knot the homotopy class of a Z[x]-complex C, such that C/(x=1) has homology Z. We'll define a metric on such Z[x]-complexes. This metric turns out to provide a lower bound (generalizing the Alishahi-Dowlin bounds) for the proper rational unknotting number, i.e. the minimal number of connectivity preserving rational tangle replacements needed to make a knot trivial. This talk is based on joint work with Damian Iltgen and Laura Marino (see https://arxiv.org/abs/2110.15107).

7- Andrew Lobb:  Squeezed knots

  • Abstract: What good are the concordance invariants arising from quantum knot homologies when compared to those arising from Floer homology?  I'll answer this question and announce a substantial prize (payable by Peter Feller) for determining whether a couple of knots satisfy an easily stated condition.  Joint work with Peter Feller and Lukas Lewark.

8- Andrew Manion: Decategorifying higher actions in Heegaard Floer homology

  • Abstract: I will discuss the decategorification of the higher actions on bordered (sutured) Heegaard Floer strands algebras arising from joint work with Raphael Rouquier (and motivated by Douglas-Manolescu's constructions in cornered Heegaard Floer homology). I will also discuss a new perspective on the sutured surfaces to which these algebras are assigned, namely the interpretation of these sutured surfaces as open-closed cobordisms, and try to explain a more flexible gluing theorem (related to open-closed TQFT) for the decategorifications of the algebras, recovering the decategorification of the cornered-Floer or higher-representation-theoretic gluing theorem as a special case.

9- Gage Martin: Annular links, double branched covers, and annular Khovanov homology

  • Abstract: 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.

10- Ina Petkova: Annular link Floer homology and gl(1|1)

  • Abstract: The Reshetikhin-Turaev construction for the quantum group U_q(gl(1|1)) sends tangles to C(q)-linear maps in such a way that a knot is sent to its Alexander polynomial. Tangle Floer homology is a combinatorial generalization of knot Floer homology which sends tangles to (homotopy equivalence classes of) bigraded dg bimodules. In earlier work with Ellis and Vertesi, we show that tangle Floer homology categorifies a Reshetikhin-Turaev invariant arising naturally in the representation theory of U_q(gl(1|1)); we further construct bimodules \E and \F corresponding to E, F in U_q(gl(1|1)) that satisfy appropriate categorified relations. After a brief summary of this earlier work, I will discuss how the horizontal trace of the \E and \F actions on tangle Floer homology gives a gl(1|1) action on annular link Floer homology that has an interpretation as a count of certain holomorphic curves. This is based on joint work in progress with Andy Manion and Mike Wong.

 

11- Radmila Sazdanovic:  Bilinear pairings on two-dimensional cobordisms and generalizations of the Deligne category

  • Abstract: The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. In this talk we will use a generalization of this approach to the Deligne category coupled with the universal construction of two-dimensional topological theories to construct their multi-parameter monoidal generalizations, one for each rational function in one variable. This talk is based on joint work with M. Khovanov.

12- Arik Wilbert: Odd Khovanov homology, odd arc algebras, and real Springer fibers

  • Abstract: Arc algebras were introduced by Khovanov in a successful attempt to lift the quantum sl2 Reshetikhin-Turaev invariant for tangles to a homological invariant. When restricted to knots and links, Khovanov's homology theory categorifies the Jones polynomial. Osváth-Rasmussen-Szabó discovered a different categorification of the Jones polynomial, called odd Khovanov homology. Recently, Naisse-Putyra were able to extend odd Khovanov homology to tangles using so-called odd arc algebras, which were originally constructed by Naisse-Vaz. The goal of this talk is to discuss a geometric approach to understanding odd arc algebras and odd Khovanov homology using Springer fibers over the real numbers. This talk reports on joint work with J. N. Eberhardt and G. Naisse.

13- Melissa Zhang: Quantum invariants and ribbon concordance, Turaev genus, and contact topology

  • Abstract: Despite differences in their foundations, Heegaard Floer homologies and quantum link homologies can give similar topological applications, due to similarities in their algebraic structures. In this talk, we will explore a few ways quantum invariants can give topological applications similar to those from Heegaard Floer homologies. In particular, we will use sl(N) web/foam homologies to give obstructions to ribbon concordance, and use spectral sequences to bound distance measurements between knots (alternation number and Turaev genus). Finally, we will discuss how to use the periodic structure of the Khovanov homology of torus knots to study contact-geometric properties of braids. This is based on joint and ongoing work with Carmen Caprau, Nicolle Gonzalez, Christine Ruey Shan Lee, Adam Lowrance, and Radmila Sazdanovic.

14- Claudius Zibrowius: Rasmussen invariants of Whitehead doubles and other satellites

  • Abstract: I will report on joint work with Lukas Lewark in which we define a concordance homomorphism from Khovanov homology that is independent of the Rasmussen invariant.  This new concordance homomorphism plays a central role in a formula for the Rasmussen invariant of Whitehead doubles and other satellites.  The proof of this formula uses the multicurve technology for Khovanov and Bar-Natan homology that I developed in previous work together with Artem Kotelskiy and Liam Watson.

 

Lightening Talks

1- Fraser Binns: Links with Khovanov homology of low rank and Rank Bounds in Knot Floer homology

  • Abstract: Zhang-Xie gave a classification of links with Khovanov homology of rank at most 8. In this talk I will discuss a different approach to their result, passing through a classification of links with Knot Floer homology of low rank via Dowlin's spectral sequence from Khovanov homology to knot Floer homology. Our key ingredient is a rank bound in knot Floer homology coming from contact topology. This is based on joint work with Subhankar Dey.

2- Emma Pickard: Non-Zero Khovanov Homology Classes

  • Abstract: A brief detailing of different forms of Non-Zero Khovanov Homology Classes and a short proof as to what makes them non-zero.

3- Mark Ronnenberg:  Associating Legendrians to Tangles via the Chern-Simons Bundle

  • Abstract: A tangle is a pair (Y,T), where Y is a 3-manifold with boundary and T is a properly embedded 1-dimensional submanifold. To every pair (Y,T), the traceless character variety functor assigns a stratified symplectic manifold and an immersed Lagrangian submanifold. The Chen-Simons bundle over the traceless character variety of a punctured surface is a principal U(1)-bundle which comes equipped with a contact 1-form on its total space. In this talk, we focus on the case of 2-stranded tangles in the 3-ball. We will briefly explain how to construct the Chern-Simons bundle, and how, given a tangle, to obtain a Legendrian submanifold in the total space of the bundle.

4- Ryan Stees: Milnor’s invariants for knots and links in closed orientable 3-manifolds.

  • Abstract: We will describe a generalization of Milnor’s invariants, originally defined for links in the 3-sphere, to knots and links in any closed orientable 3-manifold. Our invariants come in two flavors: homotopy classes which are concordance invariants, and group homology classes which are homology concordance invariants. We will also briefly relate our work to previous efforts, in particular Milnor’s classical invariants and recent homology cobordism invariants of Cha-Orr.

5- Charles Stine:  The Complexity of Shake Slice Knots

  • Abstract: Shake slice knots arise at the interface between smooth four-manifold topology and classical knot theory. The degree to which shake slice knots can fail to be smoothly slice is intimately connected to the degree to which homeomorphic smooth four-manifolds can fail to be diffeomorphic. In this talk, we will briefly summarize this connection, and then define an integer-valued measure, called 'complexity', of how far a shake slice knot is from being slice. We will then give an explicit construction of a family of shake slice knots whose members realize all possible complexities.