Seminars (HTL)

After continuing with some basics of Heegaard Floer homology, we will discuss knot Floer homology and its relation to the Heegaard Floer homology of surgeries on knots.

The main topic of this lecture series will be the Artin braid group B_n.  We study this group using two main tools:
1) linear 2-representations of B_n, and
2) annular Khovanov homology (see Eli Grigsby's lectures).
The first lecture will introduce the braid group, explain what it means for B_n to act on a category, and motivate the study of 2-representations of braid groups.
In later lectures, we will study a particular example of a (faithful) 2-representation of B_n - the categorified Burau representation - in detail, and explain how various structures of interest in the study of the braid group can be studied through the lens of this 2-representation.  Time permitting, we will discuss the relationship between linear 2-representations of B_n and geometric representation theory.

We will introduce bordered Floer homology, including some of the underlying algebraic concepts (e.g. A_\infty modules).

In this 3-lecture series we will explore the many ways in which key  representation-theoretic features of the annular Khovanov-Lee homology of braid closures give information about the surfaces they bound in the 4-ball as well as their dynamics when viewed as mapping classes of the punctured disk.

In the first lecture, we will introduce Plamenevskaya's transverse invariant in Khovanov homology and show how it can be used to detect the trivial braid (j. work with J. Baldwin). The proof uses fundamental properties of the left-invariant order on the braid group as well as algebraic properties of the Khovanov complex of braid closures.

We will continue the discussion of bordered Heegaard Floer homology, including two different perspectives on its relation to knot Floer homology.

By endowing the annular Khovanov-Lee complex with the structure of a bifiltered complex, we define a family of annular Rasmussen invariants that gives information both about the positivity of braids viewed as mapping classes and the complexity of braided surfaces bounded by their closures (j. work with T. Licata and S. Wehrli).

I'll discuss the HOMFLY-PT polynomial and its categorification (due to Khovanov and Rozansky) and explain how it ties in with some of the things Tony and Eli have been discussing.

Satellites by torus links provide multi-component versions of cables.  I will describe the region of L-space surgeries on any torus-link satellite of any L-space knot, with a result that precisely extends Hedden’s and Hom’s analogous result for cables.  More generally, I will characterize the region of L-space surgeries for iterated torus-link satellites, including the case of algebraic links.  This is joint work with Maciej Borodzik.

Co-author: Jessica Purcell (Monash U. Melbourne Australia)

In this paper we prove that if a knot or link has a sufficiently complicated plat projection, then that plat projection is unique. More precisely, if a knot or link has a 2m-plat projection, where m is at least 3, each twist region of the plat contains at least three crossings, and n, the length of the plat, satisfies n > 4m(m − 2), then such a projection is unique up to obvious rotations. In particular, this projection gives a canonical form for such knots and links, and thus provides a classification of these links.

Decorated knot cobordisms functorially induce maps on knot Floer homology.
We compute these maps for elementary cobordisms, and hence give a formula for 
the Alexander and Maslov grading shifts. We also show a non-vanishing result in the case of
concordances and present some applications to invertible concordances. 
This is joint work with Marco Marengon.

Co-author: Faramarz Vafaee (Caltech)
 
Let   be a knot in a rational homology sphere  . Then there is a unique surgery on   which results a manifold with    . We call this surgery the null surgery. When   is an L-space, the null surgery remembers the information about the genus and fiberedness of the knot. A special case of our theorem is that if the resulting manifold is      , then the dual knot is a spherical braid. This is joint work with Faramarz Vafaee.

Let M be a 3–dimensional handlebody of genus g > 1. We give examples of hyperbolic knots in M with arbitrarily large genus g bridge number which admit Dehn surgeries which are boundary- reducible manifolds.

Co-authors: Michel Boileau (Université Aix-Marseille), Cameron McA. Gordon (University of Texas at Austin)
 
We show that if L is an oriented non-trivial strongly quasipositive link or an oriented quasipositive link which does not bound a smooth planar surface in the 4-ball, then the Alexander polynomial and signature function of L determine an integer n(L) such that \Sigma_n(L), the n-fold cyclic cover of S^3 branched over L, is not an L-space for n > n(L). If K is a strongly quasipositive knot with monic Alexander polynomial such as an L-space knot, we show that \Sigma_n(K) is not an L-space for n \geq 6 and that the Alexander polynomial of K is a non-trivial product of cyclotomic polynomials if \Sigma_n(K) is an L-space for some n = 2, 3, 4, 5. Our results allow us to calculate the smooth and topological 4-ball genera of, for instance, quasi-alternating oriented quasipositive links. They also allow us to classify strongly quasipositive 3-strand pretzel knots. 

We say that   is a characterizing slope for a knot   in the 3-sphere if the oriented homeomorphism type of  -surgery is sufficient to determine the knot   uniquely. I will discuss the problem of determining which slopes are characterizing for torus knots, paying particular attention to non-integer slopes. This problem is related to determining which knots in     have Seifert fibered surgeries.

Co-author: Tye Lidman (North Carolina State)
 
I will discuss a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, it follows that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/p-L-space (for p prime), then Y is a Z/p-L-space. Further, we obtain constraints on surgeries on a knot being regular covers over other surgeries on the same knot, and over surgeries on other knots. This is joint work with Tye Lidman.

The knot concordance group C consists of knots in S^3 modulo knots that bound smooth disks in B^4. We consider C_Z, the group of knots in homology spheres that bound homology balls modulo knots that bound smooth disks in a homology ball. Matsumoto asked if the natural map from C to C_Z is an isomorphism. Adam Levine answered this question in the negative by showing the map is not surjective. We show that the image of C in C_Z is of infinite index. This is joint work with Adam Levine and Tye Lidman.

Co-authors: Cagatay Kutluhan ( University at Buffalo), Jeremy Van Horn-Morris (University of Arkansas), Andy Wand (University of Glasgow)

In this joint work with Kutluhan, Van Horn-Morris and Wand, we define the {\it spectral order} invariant of contact structures in dimension three by refining the contact invariant from Heegaard Floer homology. This invariant takes values in the set \mathbb{Z}_{\geq0}\cup\{\infty\}. It is zero for overtwisted contact structures, \infty for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. It gives a criterion for tightness of a contact structure stronger than that given by the Heegaard Floer contact invariant, and an obstruction to existence of Stein cobordisms between contact 3-manifolds. We show this by exhibiting an infinite family of examples with vanishing Heegaard Floer contact invariant on which our invariant assumes an unbounded sequence of finite and non-zero values.

What can you say about a knot's slice genus by just looking at its Seifert form?  The talk will recall the (known) answer in the smooth setting and then focus on the topological setting. The talk is based on joint work with Peter Feller.

A Weinstein manifold is an open symplectic manifold admitting a handle decomposition adapted to the symplectic structure. It turns out that the handles of such a decomposition have index at most half of the dimension. When the index is half the dimension, they are called critical handles and their cocores are Lagrangian discs.

In a joint work with Baptiste Chantraine, Georgios Dimitroglou Rizell and Roman Golovko, we decompose any object in the wrapped Fukaya category of a Weinstein manifold as a twisted complex built from the cocores of the critical handles in a Weinstein handle decomposition. The main tools used are the Floer homology theories of exact Lagrangian immersions, of exact Lagrangian cobordisms in the SFT sense (i.e. between Legendrians), as well as relations between these theories.

Since most participants of the HTL program are not experts in Fukaya categories (including me, actually) I will try to take it easy.

A 3-manifold is called an L-space if its Heegaard Floer homology has minimal possible rank. A link (or knot) is called an L-space link if all sufficiently large surgeries of the three-sphere along its components are L-spaces. It is well known that the set of L-space surgeries for a nontrivial L-space knot is a half-line. Quite surprisingly, even for links with 2 components this set could have a complicated structure. I will prove that for "most" L-space links (in particular, for most algebraic links) this set is bounded from below, and show some nontrivial examples where it is unbounded. This is a joint work with Andras Nemethi.

In this talk I review the recent progress made in defining homological invariants for 3-manifold using string theory constructions.  This generalizes the constructions of homological invariants for knots using M5 branes, to the case of 3-manifolds.

I will report on recent progress on the Gopakumar--Ooguri--Vafa correspondence, relating quantum (Witten--Reshetikhin--Turaev) invariants of 3-manifolds and knots therein with curve-counting theories (Gromov--Witten/Donaldson--Thomas) of local Calabi--Yau threefolds, in the context of Seifert-fibred 3-manifolds. I will describe A- and B- model constructions for the correspondence in the broadest context where the standard form of the duality is expect to hold (spherical space forms), discuss the link with relativistic integrable systems and the Eynard--Orantin topological recursion, and present a rigorous proof of the B-model version of the correspondence via matrix model techniques. Implications for refined invariants, orbifold GW theory, and an allied class of Frobenius manifolds and 2D-Toda reductions will be also discussed time permitting.

Based on joint work with G. Borot and further work in progress.

In this talk we will present a Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. We further present a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. We will explain how all these dimensions fit into a one parameter family of 2D TQFT's, encoded in a one parameter family of Frobenius algebras, which we will construct.​

Co-author: Piotr Sulkowski (University of Warsaw, Caltech)

In this talk I will introduce a new class of algebraic curves called extremal A-polynomials of a knot and use it to describe BPS invariants introduced by Labastida, Marino, Ooguri, and Vafa.

I will present results obtained from the analysis of both classical and quantum extremal A-polynomials. The first lead to exact formulas for BPS invariants imposing nontrivial integrality statements in number theory. The latter enabled us to construct the combinatorial model for calculating BPS invariants.

I will also indicate how these results relate to the formalism of quivers introduced in the talk by Piotr Sulkowski.

I will talk about recent progress in understanding the structure of type A link homologies. This includes the definition of integral, equivariant, colored sl(N) Khovanov-Rozansky link homologies, which are functorial under link cobordisms, as well as a study of their deformations and stability properties. I will finish by discussing some implications for colored, triply-graded HOMFLY-PT homologies, including an exponential growth property conjectured by Gorsky, Gukov and Stosic.

We present a conjectural description of Legendrian symplectic field theory for the conormal of a knot ("higher genus knot contact homology") and discuss its relation to the recursion relation for the colored HOMFLY polynomial. This reports on joint work with Lenny Ng. 

I will present a surprising relation between knot invariants and quiver representation theory, motivated by various string theory constructions involving BPS states. Consequences of this relation include the proof of the famous Labastida-Marino-Ooguri-Vafa conjecture, explicit (and unknown before) formulas for colored HOMFLY polynomials for various knots, new viewpoint on knot homologies, new dualities between quivers, and many others.

We will discuss a subclass of BPS states in the M5-brane theory on T^3 x R^3 which  are related to little strings and whose degeneracies can be worked out exactly. The generating function of these BPS states has interesting modular properties and seems to have the structure expected of the partition function with target space a symmetric product.

The goal of this talk will be to describe a rather basic linear action of the free group on a finitely-generated triangulated category, and describe two proofs of the faithfulness of this action.  I will also describe some of the geometric group theory and topology that is visible from the lens of this faithful 2-representation.

Knot Floer homology is an invariant for knots and links defined by Ozsv\'ath and Szab\'o and independently by Rasmussen. It has proven to be a powerful invariant e.g. in computing the genus of a knot, or determining whether a knot is fibered. In this talk I define a generalisation of knot Floer homology for tangles; Tangle Floer homology is an invariant of tangles in D^3, $S^2xI or in S^3. Tangle Floer homology satisfies a gluing theorem and its version in S^3 gives back a stabilisation of knot Floer homology. Finally, I will discuss how to see tangle Floer homology as a categorification of the Reshetikhin-Turaev invariant for gl(1|1).

This is a joint work with I. Petkova and A. P. Ellis.

I will outline a technique, introduced by me and Ben Elias, for computing the HOMFLY version of Khovanov-Rozansky homology.  I will show how this technique gives a simple recursion which computes the homology of positive torus links (recently accomplished in special cases by me, and in generality by Anton Mellit).  Time permitting, I would like to use these computations for 3-strand braids to illustrate a connection with recent conjectures of Gorsky-Negut-Rasmussen.

There are spectral sequences relating reduced Khovanov homology to a variety of other homological link invariants, including the Heegaard Floer homology of the branched double cover and instanton knot homology. However, there is no known relationship between Khovanov homology and knot Floer homology, despite considerable computational evidence and numerous formal similarities. I will describe ongoing efforts to find a spectral sequence relating these two invariants. Specifically, we construct a variant of Khovanov homology for links with one or more basepoints on each component, which more closely parallels the behavior of knot Floer homology and which conjecturally fits into a spectral sequence as required. This is joint work with John Baldwin and Sucharit Sarkar.

Co-author: Anthony Licata (Australian National University)

I'll begin with some dreams and motivation regarding lifting notions of lattice theory to homological algebra. For most of the talk, we'll study a concrete example: the lattices of integer cuts and flows associated to a finite graph. Given a graph G and choice of spanning tree T, we construct an algebra A(G,T) such that the Groethendieck group of the category of finitely generated A(G,T)-modules with the Euler (Ext) pairing contains the cut and flow lattices of G as orthogonal complements. We'll discus many open problems regarding generalisations and possible uses for the extra structure that is present at the category level, such as gradings. Joint work in progress with Anthony Licata.Navigation:

We'll show how categorical traces and foam categories can be used to define an invariant of braid conjugacy, which can be viewed as a "universal" type-A braid invariant. Applying various functors, we recover several known link homology theories, both for links in the solid torus, and, more-surprisingly, for links in the 3-sphere. Variations on this theme produce new annular invariants, and, conjecturally, a homology theory for links in the 3-sphere which categorifies the sl(n) link polynomial but is distinct from the Khovanov-Rozansky theory. Lurking in the background of this story is a family of current algebra representations.

This is joint work with Queffelec and Sartori.

We construct a triangulated monoidal idempotent complete category with the
Grothendieck ring naturally isomorphic to the ring of integers with two inverted.
This is a joint work with Yin Tian.

Co-author: Andrew Lobb (Durham University)

Khovanov-Rozansky homology in its most general form (so-called equivariant homology) associates to a knot a chain complex (invariant up to homotopy equivalence) over a certain polynomial ring. Equivariant homology yields various lower bounds to the slice genus, some of them concordance homomorphisms, some not; and also a piecewise linear function which has much resemblance with the recently introduced Upsilon-invariant from Heegaard-Floer homology.

I will talk about recent progress in understanding the structure of type A link homologies. This includes the definition of integral, equivariant, colored sl(N) Khovanov-Rozansky link homologies, which are functorial under link cobordisms, a study of their deformations and stability properties and the question of how these invariants extend to links in thickened surfaces. 

Joint work with M. Ehrig, H. Queffelec, D. Rose and D. Tubbenhauer.

It goes without saying that diagonalization is an important tool in linear algebra and representation theory.  In this talk I will discuss joint work with Ben Elias in which we develop a theory of diagonalization of functors, which has relevance both to higher representation theory and to categorified quantum invariants.  For most of the talk I will use small examples to illustrate of components of the theory, as well as subtleties which are not visible on the linear algebra level.  I will also state our Diagonalization Theorem which, informally, asserts that an object in a monoidal category is diagonalizable if it has enough ``eigenmaps''.  Time allowing, I will also mention our main application, which is a diagonalization of the full-twist Rouquier complexes acting on Soergel bimodules in type A.  The resulting categorical eigenprojections categorify q-deformed Young idempotents in Hecke algebras, and are also important for constructing colored link homology theories which, conjecturally, are functorial under 4-d cobordisms.  

Co-author: Mikhail Khovanov (Columbia University)

We focus on non-multiplicative TQFTs via representations of non-unital rings built out of cobordism categories modulo various relations. Examples include diagrammatic categorifications of the polynomial ring Z[x] and several classes of orthogonal polynomials.

The celebrated boson-fermion correspondence is an isomorphism between the bosonic Fock space and the fermionic Fock space. We present  categorification of the bosonic Fock space and the Heisenberg algebra which is a modification of Khovanov's Heisenberg category. The categorifcation of the fermionic Fock space is based on Honda's category studying contact topology in dimension three. This is a joint work with Mikhail Khovanov.