Abstract

We construct a spectral sequence from the Robert-Wagner gl0-homology to the knot Floer homology. This spectral sequence is of Bockstein type and comes from a subtle manipulation of coefficients. The main tools are quantum traces of foams and of singular Soergel bimodules.
This is a joint work with KRZYSZTOF K. PUTYRA, LOUIS-HADRIEN ROBERT, AND EMMANUEL WAGNER.

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Abstract

I will describe a surprising connection between the colored HOMFLYPT polynomials of knots and the motivic Donaldson-Thomas invariants of certain symmetric quivers, which was conjectured by Kucharski-Reineke-Stošić-Sułkowski. I will outline a proof of this correspondence for arborescent links via quivers associated with 4-ended tangles, which is joint work with Marko Stošić. The underlying idea is to perform web evaluation simultaneously at all (exterior) colors using generating functions. It is tempting to speculate whether this idea carries over to foam evaluation.

Abstract

In 2018, Khovanov and Robert introduced a version of Khovanov homology over a larger ground ring, termed U(1)xU(1)-equivariant Khovanov homology. This theory was also studied extensively by Taketo Sano. Ross Akhmechet was able to construct an equivariant annular Khovanov homology theory using the U(1)xU(1)-equivariant theory, while the existence of a U(2)-equivariant annular construction is still unclear.
Observing that the U(1)xU(1) complex admits two symmetric algebraic gradings, those familiar with knot Floer homology over the ring F[U,V] may naturally ask if these filtrations allow for algebraic constructions already seen in the knot Floer context, such as Ozsváth-Stipsicz-Szabó's Upsilon. In this talk, I will describe the construction and properties of such an invariant. I will also discuss some ideas on how future research might use the U(1)xU(1) framework to identify invariants similar to those constructed from knot Floer homology over F[U,V], and speculate on the topological information these constructions might illuminate.
This is based on joint work with Ross Akhmechet.

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Abstract

I will motivate the existence of homotopy types that lift link invariants. We will briefly review recent joint work with M.Khovanov on deformations of Foam evaluations using formal group laws. This deformation suggests that (complex oriented) cohomology theories seem to be making an appearance via their evaluation on spaces (or spectra) that lift Foams. We will offer some evidence that suggests that such spectra exist.

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I will present a complete foam definition of Soergel bimodules, their morphisms and their Hochschild homology.
This is a joint work with Mikhail Khovanov and Louis-Hadrien Robert.

Abstract

For a prime p, the WRT invariant of a 3-manifold lives in a cyclotomic ring. In order to categorify such rings, Khovanov developed the machinery of p-DG algebras. Building upon work of Khovanov-Rozansky, we discuss a p-DG structure on link homology. Using ideas of Cautis, Queffelec-Rose-Sartori, and Robert-Wagner, we show that it gives rise to a categorification of the Jones polynomial at a root of unity.

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Kronheimer and Mrowka recently suggested a possible approach towards a new proof of the four color theorem that does not rely on computer calculations. Their approach is based on a functor J^sharp, which they define using gauge theory, from the category of webs and foams to the category of vector spaces over the field of two elements. They also consider a possible combinatorial replacement J^flat for J^sharp. Of particular interest is the relationship between the dimension of J^flat(K) for a web K and the number of Tait colorings Tait(K) of K; these two numbers are known to be identical for a special class of "reducible" webs, but whether this is the case for nonreducible webs is not known. We describe a computer program that strongly constrains the possibilities for the dimension and graded dimension of J^flat(K) for a given web K, in some cases determining these quantities uniquely. We present results for a number of nonreducible example webs. For the dodecahedral web W_1 the number of Tait colorings is Tait(W_1) = 60, but our results suggest that dim J^flat(W_1) = 58.

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Abstract

Crane-Yetter model is expected to be a fully-extended topological quantum field theory that categorifies the Jones polynomial. We will present its categorical values for the spaces of (co)dimension 2.

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I will define an extended Crane-Yetter TQFT using skeins. In particular, given a 4D cobordism with corners, I define a map between skein modules based on a handle decomposition. The Witten-Reshetikhin-Turaev TQFT naturally appears as a boundary theory to the extended CY TQFT.

Abstract

In earlier work by Ellis, Petkova, and Vertesi, tangle Floer bimodules (a combinatorial generalization of link Floer homology) are shown to decategorify to the Reshetikhin–Turaev invariants arising in the representation theory of gl(1|1). In this talk, we describe how this algebraically gives rise to a gl(1|1) action on annular link Floer homology, viewed as the Hochschild homology—or horizontal trace—of a tangle Floer bimodule. The gl(1|1) action turns out to have an interpretation as a known basepoint action in the holomorphic Floer theory. This is based on joint work in progress with Andy Manion and Ina Petkova.

Abstract

I will explain how patched surfaces with defect circles and foams relate to separable field extensions and Galois theory, and describe a connection between overlapping foams and Sylvester double sums. I will also compare traces in two-dimensional TQFTs coming from matrix factorizations with those in field extensions.

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Abstract

An interesting aspect of Khovanov homology is that it often behaves differently when coefficients are taken in a ring of characteristic 2. I'll explain a generalization of one instance of this phenomenon to sl(P) link homology in characteristic P when P is prime. The proof uses an operator defined on sl(N) link homology for any N when coefficients are taken in a ring whose characteristic divides N.

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In a joint work with A. Oblomkov we study how link homology is related to 2-categories associated with symplectic varieties: a `commuting variety’ and a Hilbert scheme of points on C^2. I will explain the basics of our construction and its relation to the annular link homology following the work of Rina Anno and Mina Aganagic.

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Abstract

Bott-Samelson bimodules are bimodules over a polynomial ring, whose summands are Soergel bimodules. In type A, they are commonly used in the definition of triply-graded knot homology. This polynomial ring admits an action of the lie algebra sl(2) by derivations, leading to an action on Bott-Samelson bimodules, and an action on morphisms between Bott-Samelson bimodules. The raising operator in sl(2) agrees with the differential used when equipping these categories with p-dg structures. A major open question is whether this leads to a consistent action of sl(2) on Soergel bimodules, as the idempotents used to project to these summands are not invariant under sl(2). If so, this has a number of interesting implications.

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