Quantum invariants and low-dimensional topology: Link homologies and categorification

$\mathfrak{sl}_n$-homology theories obstruct ribbon concordance- Nicolle Gonzalez (University of California-Los Angeles)

Abstract: In a recent result, Zemke showed that a ribbon concordance between two knots induces an injective map between their corresponding knot Floer homology. Shortly after, Levine and Zemke proved the analogous result for ribbon concordances between links and their Khovanov homology. In this talk I will explain joint work with Caprau-Lee-Lowrance-Sazdanovic and Zhang where we generalize this construction further to show that a link ribbon concordance induces injective maps between $\mathfrak{sl}_n$-homology theories for all $n \geq 2$.


Khovanov-Rozansky homology of torus links (and beyond)- Matthew Hogancamp (Northeastern University)


Abstract: Throughout the past decade, torus links and their homological invariants have been the subject of numerous fascinating conjectures connecting link homology to seemingly distant areas of algebraic geometry and and representation theory (see work of various subsets of Cherednik, Gorsky, Negut, Oblomkov, Rasmussen, Shende). Many of these conjectures are now proven by "computing both sides" (the link homology side this was done by myself and Anton Mellit, both independently and jointly). In this talk I will discuss the main technique for computing (by hand!) Khovanov-Rozansky's HOMFLY-PT homology introduced by myself and Ben Elias, and its application to link homology. The message I hope to convey is that this technique is strikingly simple to use, and is useful in a wide variety of settings.


Khovanov homology detects T(2,6)- Gage Martin (Boston College)



Abstract: Khovanov homology is a combinatorially defined link homology theory. Due to the combinatorial definition, many topological applications of Khovanov homology arise via connections to Floer theories. A specific topological application is the question of which links Khovanov homology detects. In this talk, we will give an overview of Khovanov homology and link detection, mention some of the connections to Floer theoretic data used in detection results, and show that Khovanov homology detects the torus link T(2,6).


Generalizing Rasmussen's $s$-invariant, and applications- Michael Willis (University of California-Los Angeles)

I will discuss a method to define Khovanov and Lee homology for links $L$ in connected sums of copies of $S^1\times S^2$. This allows us to define an $s$-invariant $s(L)$ that gives genus bounds on oriented cobordisms between links. I will discuss some applications to surfaces in certain 4-manifolds, including a proof that the $s$-invariant cannot detect exotic $B^4$'s coming from Gluck twists of the standard $B^4$. All of this is joint work with Ciprian Manolescu, Marco Marengon, and Sucharit Sarkar.


19 replies on “Quantum invariants and low-dimensional topology: Link homologies and categorification”

Hello, my name is Gage Martin. I’m a graduate student at Boston College and my advisor is Eli Grigsby. If you have any questions about my talks please feel welcome to leave a comment and I will try to reply in a timely manner throughout the conference. If you’d prefer to ask me in person, my offices hour are at 1pm EST on Tuesday, June 9th joint with Elaina Aceves.

Thanks for your talk: I knew that there were some detection results like yours but now I have a much better idea of how they work.

Is there a simple way to say what the difference between Lee and Khovanov homology is? Are they both different categorifications of the Jones polynomial, or does Lee homology categorify something else?

Chain complexes for both Lee homology and Khovanov homology are built from the same underlying vector space. The differential for Lee homology can be written as the sum of the differential for Khovanov homology and some additional terms that raise the quantum grading.

Lee homology is not a categorification of the Jones polynomial. I believe the Euler characteristic of Lee homology is always 2 to the number of components of your link, so if Lee homology is viewed as categorifying something it should be something that only depends on the number of components of your link.

These are two other ways of thinking about the relationship between Khovanov and Lee homologies but if they aren’t helpful please ignore them.

If you think of Khovanov homology as “analogous” to knot Floer homology, then Lee homology should be thought of as “analogous” to the Heegaard Floer homology of your underlying 3 manifold.

I believe that you can also think of Khovanov homology as sl(2) homology where the degree 2 polynomial you use to define sl(2) homology is chosen to have a double root. Then Lee homology should instead correspond to a choice of degree 2 polynomial with two distinct roots, giving you two copies of sl(1) homology. (I am not an expert on sl(n) homology so I hope that someone corrects me if anything in this part is incorrect)

Hi Calvin,
Gage’s answer is pretty awesome for sure. But I thought I’d give an alternative perspective which was what ended up being the most helpful for my own work.

If we follow Bar-Natan’s approach to defining Khovanov homology via dotted cobordisms between planar diagrams (page 1493 in – we have a complex over the category where objects are 1-mflds in the disk, and morphisms are cobordisms in disk times [0,1] ), there is a question about what to do with multiple dots on a given connected cobordism. For Khovanov homology, we simply declare that having any more than one dot causes the entire cobordism to be equivalent to the zero map. For Lee homology, we declare that the number of dots is taken modulo 2 (or equivalently, 2 dots are equivalent to no dots). In general, 2 dots can be declared equivalent to a formal parameter $t$ which “deforms” the Khovanov complex in the sense that $t=0$ recovers Khovanov homology, while $t=1$ gives Lee homology.

The ‘bad’ news is that Lee homology (over a ring with 2 invertible) is equivalent to the number of possible orientations of your link – not very useful as an invariant or categorification of anything a priori. The ‘good’ news is that there is also a filtration on the Lee complex (a ‘deformed’ version of the grading on the Khovanov complex used to relate to the Jones polynomial) which gives rise to Rasmussen’s $s$-invariant, which gives wonderful topological information.

Michael, those are interesting results. Could you say a little more about the reason for comparing $s(L)$ to $s(F_{p,p})$, for those special links $F_{p,p}$?

Hi Carmen,
Thanks! Okay so it’s a little hard in text, but the basic idea is as follows (let’s stick with a single copy of $S^1\times S^2$, rather than a bunch of connected sums). If you have some surface $\Sigma\subset B^2\times S^2$ with $\partial\Sigma=L\subset S^1\times S^2$, you can get it transverse to the “core” $C:=\{0\}\times S^2$, but not necessarily disjoint from this core.

Now drill out a neighborhood $N(C)$ of the core, and you’ll see $\Sigma$ providing a cobordism between $L$ in the “outer boundary” and a new link $L’\subset \partial N(C) \cong S^1\times S^2$. What is this link $L’$? Well, for each point in the transverse $\Sigma \cap C$, the resulting $\Sigma\cap \partial N(C)$ will look like a small $S^1$-fiber in $\partial N(C) \cong S^1\times S^2$. Since $\Sigma$ is a cobordism from a null-homologous link, it must have cancelling pairs of such fibers – which is precisely what we’re calling $F_{p,p}$ for various $p$.

I hope this answered the question, but we can talk about it more for sure!

Gage, have you tried doing similar things for other torus links? What would the difficulties be there?

Carmen, I think that is a great question! I should point out that a consequence of recent work by Yi Xie and Boyu Zhang is that Khovanov homology detects the torus link T(3,3)( = L6n1 with the correct orientations chosen)

Seeing as how Khovanov homology detects T(2,2), T(2,4) and T(2,6) I think a natural question is what about T(2,2n) more generally. Unfortunately there are some issues with extending the proof. I think the biggest issue is that there are infinitely many conjugacy classes 4-braids which represent the unknot which is different than the case for lower braid index. I am happy to chat more about if there are other (torus) links where applying some of these techniques might be useful for showing detection results.

Hi Gage,
Great talk! So if I’m understanding you right, it’s reasonable to think that the spectral sequence arguments would work out fine (probably since $T(2,2n)$ are alternating so the Khovanov homology is nice and thinly spread out?) but at the end you would arrive at braids for the unknot of larger and larger braid index (corresponding to the linking number with the unknot component). And there are infinitely many such unknot-braids… is there any way of writing these as infinite families of finitely many types, and case-checking from there? Also, do all of these unknot-braids satisfy the property that, if you untwist them, the OTHER unknot becomes a braid (the same braid?)? I don’t know much about this, just throwing out random ideas, cause why not.

Hi Mike,

These are great questions. I think with regards to the spectral sequence arguments, some probably should work in similar ways but others will get (much) harder as the rank of Khovanov homology grows. I think it is harder to say exactly where they would get “too hard” though.

I am not aware of a way of writing the braids as of finitely many types, though that certainly doesn’t mean there isn’t one (and I would love to hear about it if there is one). I think the nicest classification I know of is through braid folliations on the disks bounded by the unknoted braid closures.

Not all of the unknot-braids satisfy the property that, if you untwist them, the OTHER unknot becomes a braid. This property is called being “exchange-braided”. Unfortunately, even in the proof the Khovanov homology detects T(2,6) I wasn’t able to show exchange braidedness, just that one component is a braid (though once you know that Kh detects T(2,6) exchange braidedness will follow).

For anyone who has watched Matt’s great talk and mine, I just wanted to mention that a large part of what I skipped in my talk was also related to homological perturbation theory. We were manipulating Khovanov and Lee complexes rather than Khovanov-Rozansky complexes, but the ideas are the same. The key difference was that, in fact, we did need to keep track of how the differentials changed (or more precisely, how they did NOT change) during our simplifications. We also needed to make sure that our homological perturbations did not shift filtration levels in an uncontrolled manner, or interfere with the interpretation of Lee homology as based solely upon orientations of the link.

Hi Nicolle!
So I’m sure this is the standard question any time you guys bring this up, but… do you have any examples of obstructing ribbon concordance between two links that aren’t also obstructed by Knot Floer?

Also, is there reason to believe in a similar result for HOMFLY homology in the limit?

Hi Mike,

I assume this is a question you get a lot but if I take an oriented link L in S^3, is there a relationship between the s-invariant of that link and the s-invariant you’ve constructed for the “knotified” link in the connect sum of some S^1 X S^2’s?

Hey Gage,
Actually, we didn’t put a whole lot of thought into knotifications, but now that you ask, I’m pretty certain it’s fairly easy to show that, if $L$ has $m$ components, then $|s(L) – s(\kappa(L))| \leq m-1$. We have a stronger conjecture in our paper about $s(L)$ and $-s(-L)$, but only a handful of cases checked and an analogy with $\tau$ to go on.

Hi Mike,
I tried to look in your paper but was having trouble finding this conjecture? Also just to make sure I understand, is this bound coming from a cobordism between L and k(L)?

Hi Gage,
It’s sort of tucked away. Knotifications are talked about in Section 8.4. The conjecture is written as Question 8.24. In that notation, $s_+(L):= -s(-L)$. And yeah, $L$ can be placed in a trivial 3-ball within $\#^r(S^1\times S^2)$, after which there is a simple cobordism to $\kappa(L)$ where we can push the relevant points of the link through the various handles before joining them via saddle cobordisms. Such a cobordism is clearly connected.

Hi Matt!
Nice talks and very interesting construction. I should read your paper. For what other links have you tried to apply this construction (or think that it would work nicely), besides positive torus links?

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