Schedule for: 21w5105 - Perspectives on Knot Homology (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Thirty years ago, work of Witten and Reshetikhin-Turaev activated the study of quantum invariants of links and three-manifolds. A cornerstone of subsequent developments, leading up to our current knot-homology conference, was a three-pronged approach involving 1) quantum field theory (Chern-Simons); 2) rational VOA's (WZW); and 3) semisimple representation theory of quantum groups. The second and third perspectives have since been extended, to logarithmic VOA's and related non-semisimple quantum-group categories. I will propose a family of 3d quantum field theories that similarly extend the first perspective to a non-semisimple (and more so, derived) regime. The 3d QFT's combine Chern-Simons theory with a topologically twisted supersymmetric theory. They support boundary VOA's whose module categories are dual to modules for Feigin-Tipunin algebras and (correspondingly) to modules for small quantum groups at even roots of unity. The QFT is also compatible with deformations by flat connections, related to the Frobenius center of quantum groups at roots of unity. This is joint work with T. Creutzig, N. Garner, and N. Geer. I will mention potential connections to related recent work of Gukov-Hsin-Nakajima-Park-Pei-Sopenko and promising routes to categorification, from a physics perspective.

I will survey link concordance invariants coming from Khovanov homology, particularly those similar in spirit to Ozsváth-Stipsicz-Szabó's Upsilon, a 1-parameter family of invariants coming from knot Floer homology. This is related to my joint work with Linh Truong on annular link concordance invariants as well as ongoing work with Ross Akhmechet.

A well-known strategy to disprove the smooth 4D Poincare conjecture is to find a knot that bounds a disk in a homotopy 4-ball but not in the standard 4-ball. Freedman, Gompf, Morrison and Walker suggested that Rasmussen’s invariant from Khovanov homology could be useful for this purpose. I will describe three recent results about this strategy: that it fails for Gluck twists (joint work with Marengon, Sarkar and Willis); that an analogue works for other 4-manifolds (joint work with Marengon and Piccirillo); and that 0-surgery homeomorphisms provide a large class of potential examples (joint work with Piccirillo).

Please turn on your cameras for the "group photo" -- a screenshot in Zoom's Gallery view.

We describe invariant counts of holomorphic curves in a Calabi-Yau 3-fold with boundary in a Lagrangian in the skein module of that Lagrangian. We show how to turn this into concrete counts for the toric brane in the resolved conifold. This leads to a notion of basic holomorphic disks for any knot conormal in the resolved conifold. These basic holomorphic disks seem to generate HOMFLY homology in the basic representation. We give a conjectural description of similar holomorphic object generating parts of higher symmetric representation HOMFLY homology and verify some predictions coming from this conjecture in examples.

In this talk, I will motivate the equations of gauge theory in four or five dimensions that can be used to give a dual description of the Jones polynomial by counting solutions of certain elliptic partial differential equations, and a construction of Khovanov homology.

We define a new family of commuting operators F_k in Khovanov-Rozansky link homology, similar to the action of tautological classes in cohomology of character varieties. We prove that F_2 satisfies "hard Lefshetz property" and hence exhibits the symmetry in Khovanov-Rozansky homology conjectured by Dunfield, Gukov and Rasmussen. This is a joint work with Matt Hogancamp and Anton Mellit.

We will discuss properties of the stable homotopy refinement of Khovanov homology and some aspects of its construction. We will focus on features that also appear for other Floer-type invariants, and on gaps in our understanding. The results are joint with Tyler Lawson and Sucharit Sarkar (or are due to other people).

Khovanov showed, more than 20 years ago, that there is a deeper theory underlying the Jones polynomial. The “knot categorification problem” is to find a uniform description of this theory, for all gauge groups, which originates from physics, or from geometry. I will describe two solutions to this problem, which I recently discovered, related by a version of two dimensional homological mirror symmetry. I explicitly solve the theory that categorifies the Jones polynomial. The result is a new geometric formulation of Khovanov homology, which generalizes to all groups.

I will discuss recent work with Raphael Rouquier, focusing on a higher tensor product operation for 2-representations of Khovanov's categorification of U(gl(1|1)^+), examples of such 2-representations that arise as strands algebras in bordered and cornered Heegaard Floer homology, and a tensor-product-based gluing formula for these 2-representations expanding on work of Douglas-Manolescu.

Foams are surfaces with singularities which can be thought of as cobordisms between graphs. Foam evaluation is a combinatorial formula which associates a symmetric polynomial to any closed foam. I will describe this combinatorial formula and explain how it can be used to construct link homology theories. Finally I will relate foam evaluation to Soergel bimodules and give a foamy description of their Hochschild homology. Joint with Mikhail Khovanov and Emmanuel Wagner.

The Khovanov complex of a link L in a thickened annulus carries a filtration; the associated graded complex gives rise to the annular Khovanov homology of L. Grigsby-Licata-Wehrli show that this annular homology admits an action by the Lie algebra sl(2). Using the techniques of Lipshitz-Sarkar, one can define a stable homotopy lift of the annular Khovanov homology of L. In this talk I will describe (in part) how to lift the sl(2)-action to the stable homotopy category as well. This is joint work with Ross Akhmechet and Slava Krushkal.

Recent work of Aganagic details the construction of a homological knot invariant categorifying the Reshetikhin-Turaev invariants of miniscule representations of type ADE Lie algebras, using the geometry and physics of coherent sheaves on a space which one can alternately describe as a resolved slice in the affine Grassmannian, a space of G-monopoles with specified singularities, or as the Coulomb branch of the corresponding 3d quiver gauge theories. We give a construction of this invariant using an algebraic perspective on BFN's construction of the Coulomb branch, and in fact extend it to an invariant of annular knots. This depends on the theory of line operators in the corresponding quiver gauge theory and their relationship to non-commutative resolutions of these varieties (generalizing Bezrukavnikov's non-commutative Springer resolution).

Ribbon categories are 3-dimensional algebraic structures that control quantum link polynomials and that give rise to 3-manifold invariants known as skein modules. I will describe how to use Khovanov-Rozansky link homology, a categorification of the gl(N) quantum link polynomial, to obtain a 4-dimensional algebraic structure that gives rise to vector space-valued invariants of smooth 4-manifolds. The technical heart of this construction is the functoriality of Khovanov-Rozansky homology in the 3-sphere. Based on joint work with Scott Morrison and Kevin Walker.

What do annular Khovanov homology, Ozsvath-Szabo's "correction terms", Kapustin-Witten equations, and enumerative BPS invariants have in common? The goal of the talk will be to explain, from multiple perspectives, how this structure makes a somewhat surprising appearance in a problem of generalizing Khovanov homology to homology of knots in arbitrary 3-manifolds.

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.