Quantum invariants and low-dimensional topology: New quantum invariants and other connections

Sequence of Undetectable Nonquasipositive Braids - Elaina Aceves (University of Iowa)

When $K$ is a quasipositive transverse knot, Hedden and Plamenevskaya proved that $2 \tau(K)-1=sl(K)$, where $\tau$ is the Ozsv\'ath-Szab\'o concordance invariant and $sl$ is the self-linking number of the knot. We construct a sequence of braids whose closures are nonquasipositive knots where we add more negative crossings as we progress through the sequence. Therefore, the knots formed from the sequence become more nonquasipositive as the sequence progresses. Furthermore, we conjecture that the knots obtained from this sequence behave like quasipositive knots in that they uphold the equality $2 \tau(K)-1=sl(K)$. Because of this property, the nonquasipositive knots obtained from our sequence are undetectable by transverse invariants like Ozsv\'ath and Szab\'o's $\hat{\theta}(K)$ from Heegaard Floer homology and Plamenevskaya's $\psi(K)$ from Khovanov homology.


On framing changes of links in $3$-manifolds- Dionne Ibarra (George Washington University)

Abstract: We show that the only way of changing the framing of a link by ambient isotopy in an oriented $3$-manifold is when the manifold admits a properly embedded non-separating $S^2$. We will illustrate the change in framing by the Dirac trick then relate the results to the framing skein module. Coauthors: Rhea Palak Bakshi, Gabriel Montoya-Vega, Jozef Przytycki, and Deborah Weeks.


Holonomy invariants of links and Holonomy invariants from quantum sl_2 and torsions- Calvin McPhail-Snyder (University of California-Berkeley)

Holonomy invariants of links

Holonomy invariants from quantum sl_2 and torsions

Abstract: To describe geometric information about a space X, we can equip it with a representation \rho: \pi_1(X) \to G, where G is a group. Quantum invariants using this extra data are called quantum holonomy invariants or homotopy quantum field theories. In this talk, I will give some motivation for this idea (including connections to volume conjectures) and discuss how to modify the usual Reshetikhin-Turaev construction to get holonomy invariants of links in S^3.

Blanchet, Geer, Patureau-Mirand, and Reshetikhin have constructed a holonomy invariant for G = SL_2(C) using the quantum group U_q(sl_2) at a root of unity. In this talk, I will give an overview of their construction, then discuss my recent work showing how to interpret their invariant in terms of twisted Reidemeister torsion.


How to make quantum groups easier? and Quantum knot invariants according to Alexander- Roland van der Veen (University of Groningen)

Joint work with Dror Bar-Natan (Toronto)

How to make quantum groups easier?

Abstract: Quantum groups such as U_q sl_2 often appear at the foundations of most quantum knot invariants. And usually this results in forbidding lists of generators, relations and formulas. Does it have to be this way? One way out is to pass to representations but then all computations will usually grow exponentially. (Try computing the Jones polynomial of a 50 crossing knot). In this talk I would like to propose another way out. Will modify the algebra itself and then work with it through Gaussian expressions. In fact one can also understand much of the way quantum groups are built from a purely topological standpoint but that is the subject of another talk.

Quantum knot invariants according to Alexander

Abstract: From the point of view of universal invariants the Alexander polynomial is the most fundamental quantum invariant. It appears as the one loop contribution in the perturbative expansion of the Chern-Simons integral regardless of what gauge group one starts with. This suggests that one should be able to use Alexander to gain insight into more complicated quantum invariants and conversely presents the challenge of generalizing the many nice properties of Alexander to a wider context. I will make these points concrete by reproving and extending the formula for the Alexander polynomial from a Seifert surface towards the universal quantum sl_2 invariant.


17 replies on “Quantum invariants and low-dimensional topology: New quantum invariants and other connections”

Hi everyone my name is Roland and I enjoyed watching your talks and hope we can chat about them and related quantum invariant stuff here. You can find some more about me on the Introductions page and my website is here:
I am hosting office hours today June 2 at 4pm on zoom (see Live Events) in case you want to know more about what I’m doing and/or what I talked about here.

I had one question about your talks that other people might be interested in, but I’ll probably stop by your office hours as well:

I assume the OU form of a braid is not an invariant of the closure, but how badly does that fail? Can you say anything nice about the OU form of a link (in the sense of the OU form of a braid whose closure is the link) or is it hard to compare them from different braids?

Could you motivate the choice of $\epsilon$ where $\epsilon^2 = 0$? I assume it has something to do with the resulting nice form of $Z$, and just possibly the simplest choice one can make to simplify the expansion of the exponential. But presumably other choices of power $\kappa$ where $\epsilon^{\kappa} = 0$ would also yield nice formulas?

Epsilon was introduced into the quantum group as a parameter and once it’s there it is also in all knot invariants that come out so then you can do a Taylor series in epsilon (around 0 say) to get simpler invariants. Setting $\epsilon^2=0$ just picks up the terms linear in $\epsilon$. As the constant term $\epsilon=0$ yields Alexander the terms proportional to $\epsilon$ represent the first interesting bit. More complicated than Alexander, vastly simpler than Jones, yet still strong enough to separate all knots in the Rolfsen table.

I added some bonus material to my talk explaining some more about OU tangles, Drinfeld double and a Mathematica file that actually computes some of the invariants I am talking about. All of this is work in progress.

Good Afternoon,
My name is Dionne Ibarra I am a Ph.D. student at the George Washington University. I have enjoyed watching your talks and welcome any questions and discussions.

Hi Elaina, I enjoyed your talk! Can you elaborate on how to show that the braids $K_n$ are non-quasipositive? Does growing more non-quasipositive mean that the braids are getting more crossings away from a quasi-positive knot?

Hi Linh and thanks for watching my talk! Great questions! Yes, I use the phrase “growing more non-quasipositive” to say that the braid/knot is getting more negative crossings away from a quasipositive braid/knot. For example, in “Positivities of knots and links and the defect of Bennequin inequality” by Jesse Hamer, Tetsuya Ito, and Keiko Kawamuro they define a braid as being “almost quasipositive” if it contains a quasipositive braid word but with exactly one negative crossing added to the braid word making it nonquasipositive. Using KnotInfo I was able to discover that K_1 in my sequence is nonquasipositive. Since I only add negative crossings to the braids to progress in my sequence, they grow “more nonquasipositive” as the sequence progresses.

Hi Elaina,

Cool talk! I don’t really know anything about quasipositive braids/knots (beyond what you’ve just told me) so I have a few “basic” questions:

(1) What are some motivations for wondering if a braid/knot is quasipositive or not? Is the result by Hedden and Plamenevskaya one reason why someone would want to know if a know is quasipositive, or is it used more often as a tool to determine nonquasipositiveness?

(2) How easy or hard is it to see whether a braid or knot is quasipositive from its diagram? (Well, I assume probably very hard… I was just looking at your example braid and seeing how the diagram corresponded to the braid word, and saw that you can kind of find the “positive crossings” and then see that around each one the nearby strands/crossings are “mirrored,” coming from the conjugation. But if a braid is not in this form already, I see that it would probably be tough to put it in this form.)

(3) Is there some kind of measure of “how nonquasipositive” a braid/knot is?

Hi Sarah! Thanks for listening to my talk and thanks for your excellent questions!

1. My understanding of why quasipositive braids/knots are interesting is that they “behave nicely”. The example I used in the talk was that quasipositive knots have the property that $sl(K) = 2 \tau(K)-1$ when for any knot we can only say that $sl(K)=\tau(K)-1$. Another example is Rasmussen’s concordance invariant $s$. For any knot, we know that $sl(K) \leq s(K)=1$ but for quasipositive knots we know that $sl(K)=s(K)-1$. In regards to your other question, I have been using the result to determine if a knot is quasipositive or not. We know that a quasipositive knot results in a nonzero result for both invariants, so if we obtain a trivial answer then we know the knot is nonquasipositive. Quasipositive braids are good to look at when trying to prove a result as well. If you can prove something in the case of quasipositive braids, you might be able to generalize it to all braids.
2. This is usually a difficult problem. If you can find the quasipositive braid word, great! However, if you struggle for a while and can’t find the quasipositive braid word, is it because the braid is nonquasipositive or have you just not found it yet? That’s why the two knots I mentioned in the beginning of my talk (12n239 and 12n512) are causing problems; we suspect they are nonquasipositive but the invariants we have at our disposal haven’t been able to confirm this yet.

A quick Q: when you say

“The example I used in the talk was that quasipositive knots have the property that (…) when for any knot we can only say that (…)”

Should there be an inequality there for the “any knot” part?

Yes, there should be. Sorry about that. I can’t edit the comment after I post it so I couldn’t fix it. It should read “The example I used in the talk was that quasipositive knots have the property that $sl(K)=2 \tau(K)-1$ when for any knot we can only say that $sl(K) \leq 2 \tau(K)-1$” .

Hi Dionne! Is it knows whether the $q$-homology skein module admits some additional structure?

Hi Elaina, Dionne, Calvin, and Roland!
I enjoyed your talks very much, and have learned from them too!

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