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# Investigating the L-space conjecture: With Dehn surgery

L-space knots do not have essential Conway spheres- Tye Lidman (NC State)

Abstract: The properties of a knot are heavily governed by the essential surfaces that sit in the exterior. We will study a relation between essential planar surfaces in a knot exterior and knot Floer homology. This is joint work with Allison Moore (VCU) and Claudius Zibrowius (UBC).

Introduction to L-space links- Beibei Liu (Bonn)

Abstract: L-spaces are simplest 3-manifolds in terms of Heegaard Floer homology and L-space links are links such that all large surgeries are L-spaces. In this talk, we will concentrate 2-component L-space links which is a family of simple” links in the sense that their Alexander polynomials contain full information of the link Floer complex, and give explicit answers to questions relating to the link itself and its surgeries such as some detection results, sharp slice genus bounds and Thurston polytope.

Fibred knots, positivity and L-spaces- Filep Misev (Regensburg)

Abstract: Torus knots are lens space knots: they admit surgeries to lens spaces. This classical theorem has a modern analogue in terms of Floer homology: algebraic knots are L-space knots. I will present knots which do not admit L-space surgeries despite strikingly resembling algebraic knots and L-space knots in general. More precisely, we will see a method which allows to construct infinite families of knots of arbitrary fixed genus g > 1 which are all algebraically concordant to the torus knot T(2,2g+1) of the same genus and which are fibred and strongly quasipositive. Besides the study of L-spaces, these knots are of interest in the context of knot concordance, in particular Fox's slice-ribbon question, as well as Boileau-Rudolph's question, or Baker's conjecture, on the independence of strongly quasipositive fibred knots in the concordance group. Joint work with Gilberto Spano.

Non-left-orderable surgeries on iterated 1-bridge braids- Zipei Nie (Princeton)

Abstract: We prove that the L-space conjecture holds for those L-spaces obtained from Dehn surgery on knots which are closures of iterated 1-bridge braids, i.e., the braids obtained from satellite operations on 1-bridge braids. In the proof, we emphasize the power of fixed points in the Homeo_+(R) representation, and introduce property (D) to handle the satellite operation.

## 22 replies on “Investigating the L-space conjecture: With Dehn surgery”

[…] … with Dehn surgery […]

TyeLidmansays:

Hi NCNGT conferencegoers,

I’m Tye Lidman and I am an assistant professor at North Carolina State University. I’m interested in connections between low-dimensional topology and Floer homology, including the L-space conjecture. I’m looking forward to meeting lots of new people and talking some math, so please ask questions – they don’t have to be about my talk – or just introduce yourself!

Allison Millersays:

Hi Tye! Thank you for a lovely talk- I really enjoyed both pieces! Can you say a little more (or give a reference) about the idea that for prime 3-manifolds bounding a symplectic manifold with $b_2^+>0$ is conjecturally the same as not being an L-space? (Is this the same as LO $\pi_1$ or COTF-having for some reason?)

TyeLidmansays:

Hi Allison,

I don’t think the equivalence of this conjecture is written down explicitly, but the idea is as follows. If a 3-manifold bounds such a symplectic 4-manifold, it is not an L-space (see e.g. Theorem 1.4 in https://arxiv.org/pdf/math/0311496.pdf). On the other hand, if a prime 3-manifold Y is not an L-space, by the L-space conjecture it has a taut foliation. One can then apply the Eliashberg-Thurston construction to put contact structures on Y and -Y such that Y x I provides a symplectic filling. By a theorem of Eliashberg or Etnyre (https://arxiv.org/pdf/math/0311459.pdf or https://arxiv.org/abs/math/0312091), this can be capped off to a closed symplectic 4-manifold, and further this can be chosen so that each of the two components of the cap has b^+ > 0 (Proposition 15 of https://arxiv.org/pdf/math/0311489.pdf). Hence, this provides symplectic fillings for Y and -Y with b^+ > 0.

Joao.Nogueirasays:

Hi, Tye,
Thank you for your talk. I enjoyed both parts very much!
What else is known regarding other essential surfaces in the knot exterior being an obstruction for L-space surgeries on the knot? (Besides the satellite torus from a connected sum and the essential Conway sphere.)

TyeLidmansays:

Hi Joao,

The only other general obstruction that comes to mind for hyperbolic L-space knots is that there are no properly embedded punctured projective planes (Corollary 1.5 in https://arxiv.org/pdf/1307.5317.pdf). There is also Ken Baker’s result (referenced in a response to Ian’s question below) which shows that there can be closed incompressible surfaces of arbitrarily large genus in the exteriors of knots with lens space surgeries.

Filip.Misevsays:

Good morning!
Filip Misev here; I’m in Regensburg right now and looking forward to learning something about L-spaces and the knots in your head (and the ones outside as well). Don’t hesitate to ask me and let me know about the math (and, more generally, geometry) you’re thinking about.

HannahTurnersays:

For Zipei:

Can you say anything about possible generalizations of your argument that the manifold v2503 has all non left-orderable surgeries? Are there any other 3-manifolds with torus boundary and fundamental group generated by two generators all of whose (non-longitudinal) surgeries are L-spaces for which you have/could check whether the arc of your argument runs? Or could your argument be modified for 3-manifolds (with torus boundary, two-generator fundamental group) but which have only an interval of L-space slopes?

Zipei_Niesays:

Hi Hannah,

For $v2503$, we can show that, any homomorphism $\rho$ from $\pi_1(v2503)$ to $\mbox{Homeo}_+(\mathbf{R})$, every fixed point of $\rho(\mu)$ is a global fixed point, and that $\rho(\lambda)$ have fixed points. Since after any Dehn surgery $\mu^p \lambda^q=1$ with nonzero slope, a fixed point of $\rho(\lambda)$ is a fixed point of $\rho(\mu)$, we reach a contradiction. Zero slope case can be done separately.

Zipei_Niesays:

We can definitely try more examples, and they do not have to be generated by two elements. Here is why. The only benefit of having two generators is that we could have a higher chance of getting a relation generated by $x$, $x^{-1}$ and $y$, so that we know a fixed point of $\rho(x)$ is a fixed point of $\rho(y)$. However, even this relation exists, it does not necessarily show up free of charge. In the two-generator, two-relation group, our desired relation is generated by two relations up to conjugation, inverse and multiplication, which have many possibilities. A better way to transfer the fixed points, is to find subwords in common in the two relations. Say, if we have relations $r_1=w_1 s w_2 s w_3$ and $r_2=w_4 s^{-1} w_5 s^{-1} w_6 s^{-1} w_7$ (totally made up), where $w_i (1\le i \le 7)$ are generated by $x$, $x^{-1}$ and $y$, and $s$ is the constructed subword, then a fixed point of $\rho(x)$ is a fixed point of $\rho(y)$. As we can see in this example, having two generators is only harmful, because it hides the important subword $s$.

I am writing a computer program to generate more examples of $(1,1)$ L-space knots and to compute their fundamental groups. If you can come up with examples of $3$-manifolds with torus boundary and L-space fillings, that would be nice.

HannahTurnersays:

Lidman and Watson describe infinitely many (non-hyperbolic) three-manifolds with all (non-longitudinal) L-space fillings:

https://arxiv.org/pdf/1208.3917.pdf

These surgeries might already be known to be non-LO for other (perhaps obvious) reasons, but it would be interesting if your dynamical approach works here.

Also Gillespie comes up with a topological characterization of a manifold with torus boundary with all (non-longitudinal) L-space fillings and gives some more hyperbolic examples here:

https://arxiv.org/pdf/1603.05016.pdf

I don’t know how hard it is to come up with a presentation for the fundamental group of any of these manifolds, but it would be very interesting if you could show that all of their fillings are non-LO.

SiddhiKrishnasays:

Hi Filip,

Thanks for two great talks! I really enjoyed them.

Here is a question: the knots $K_n$ seem complicated, even for $n=1$. In your paper, you mention that most (probably all) of your knots cannot be realized by positive braid closures. Is it possible to find the SQP braids whose closures are these knots? I’m guessing, in a similar vein, that the braid indices must be getting arbitrarily large, is that right?

Thanks again!

Filip.Misevsays:

Hi Siddhi,

Thanks!
Yes, that’s right; although they are relatively simple in terms of the monodromy, the knots quickly become complicated in terms of crossing number, and the (sqp) braid indices have to grow.
This is because for a given (sqp) braid index and fixed genus, the number of sqp braids is finite (the Euler characteristic of the sqp surface is #(braid strands)-#(positive half-twisted bands in the sqp braid), which is constant for fixed g. So you also get a fixed number of half-twisted bands – and the combinatorics becomes finite (ask me again if unclear)).
Also note that since g_4=g for these knots, this also has implications about concordance (e.g. a concordance class contains at most finitely many such knots when you bound the braid index).

I had drawn the braid for K_1 some time ago and if I remember correctly, it had 15(?) braid strands. To find it, first make a huge drawing of the T(2,4) torus link as a positive braid (easy, given enough paper, or a blackboard), then plumb the last, complicated Hopf band in this picture. Finally use the constructive proof of Rudolph’s result (“Quasipositive plumbing”) to make it a sqp braid. It helps not to actually draw the surface, but only the corresponding “fence diagram” (see Rudolph).

I didn’t try to find a formula for a sqp braid word for K_n; might be feasible, but certainly will take time.
Let me know if something is unclear (my answer here is a bit approximative, and I could also for example try to reconstruct the sqp braid for K_1 and send it by e-mail, if you want).

Greetings from Regensburg!
Filip

SiddhiKrishnasays:

Hey Tye! Great talks, I really enjoyed both of them.

I also have a question: can you say a bit more about what you mean by “reparametrization”? You said something like, “up to reparametrization, all rational tangles curves and irrational tangle curves have the following immersed curve representatives”. But I think I don’t exactly know what reparametrization means? Is it easy to say in a sentence or two?

Thanks again!

TyeLidmansays:

Hi Siddhi,

By reparameterization, I really just mean after applying an element of the mapping class group of the 4-punctured sphere. For context, there is a QP^1’s worth of rational tangles. However, they are all equivalent, just not rel boundary. We can untwist a rational tangle until it becomes very simple. Similarly, for each fixed component of HFT, up to an analogous automorphism of the 4-punctured sphere, one can turn the curve into one of the ones given. (One may need to apply different mapping classes for different components of HFT of a fixed tangle though, such as for the (2,-3)-pretzel tangle.)

Best,
Tye

claudius_zibrowiussays:

Hi Siddhi,

There are at least two different ways of defining Conway tangles: There is the topological definition (embeddings $latex I\cup I\hookrightarrow B^3$, such that $latex \partial I\cup \partial I \hookrightarrow \partial B^3$, considered up to isotopy) and the combinatorial one (tangle diagrams, considered up to Reidemeister moves, fixing the tangle ends).

These two definitions are not equivalent: For example, the trivial tangle )( and the one-crossing tangle ⤬ are the same with respect to the first definition, but different with respect to the second. One way to make them equivalent is to slightly modify the first definition by adding a parametrization. By this, I mean an embedded circle on $\partial B^3$ (for instance a great circle) containing all four tangle ends. The corresponding tangle diagram is obtained by choosing a projection which maps the fixed circle to the boundary of the tangle diagram.

Using the modified version of the first definition, a reparametrization of a tangle just means a different choice of fixed embedded circle connecting the four tangle ends. In the second definition, this boils down to adding twists to the tangle diagrams.

Ok, that was more than two sentences; let me know if you have more questions!

HannahTurnersays:

For Beibei: Great talks! I haven’t seen this definition of strong 4-genus before for a link. Can you say why this is the kind of surface that you can handle in your argument? Could there be bounds on other variants of 4-genus for links (say the minimum over connected surfaces, or over all surfaces) coming from link floer?

Ian.Agolsays:

Hey Tye,
Do you think that Floer homology may contain other information about incompressible surfaces? E.g. can an L-space knot have non-trivial tangle decomposition (meridional incompressible planar surface)? E.g. I recall that this case of the cyclic surgery theorem was handled with different techniques than the small knot case, but I don’t recall right now exactly what restrictions were given.

TyeLidmansays:

Hi Ian,

I think that Floer homology should contain information about other incompressible surfaces in some capacity, but not really sure how to approach such a question. For example, Baker and Moore conjecture that L-space knots have no non-trivial tangle decompositions (Conjecture 19 here https://arxiv.org/pdf/1404.7585.pdf).

There were already some similar results about Heegaard Floer homology for incompressible tori in work of Hanselman, J. Rasmussen, and Watson (Section 1.6 of https://arxiv.org/pdf/1604.03466.pdf).

It’s also worth pointing out that L-space knots can have closed incompressible surfaces of higher genus in their exteriors, by Baker’s work https://arxiv.org/pdf/math/0509082.pdf. I’m not sure if the Alexander polynomials of these knots are somehow different than the knots that don’t have such surfaces.

SubhankarDeysays:

Hi Tye, loved both of your talks. Thanks a lot for those.

Is it known how knot Floer homology behaves if one replaces an essential tangle within a knot by a different one? Eg. replacing a non-nugatory crossing by a rational tangle?

TyeLidmansays:

Hi Subhankar,

A not very satisfying answer is that there are long exact sequences in knot Floer homology like the Skein relations for knot polynomials, which helps one to compute the knot Floer homology of crossing changes or smoothings (oriented – p.3 of https://arxiv.org/pdf/math/0209056.pdf, unoriented – https://arxiv.org/pdf/math/0609531.pdf or https://arxiv.org/pdf/1305.2562.pdf). For a general rational tangle replacement, if one understands HFT(T), then it is very easy to compute the knot Floer homology of any rational tangle filling using the immersed curve theory, since one just intersects HFT(T) with an embedded curve in the 4-punctured sphere with slope specified by the rational tangle.

TyeLidmansays:

Hi Zipei,

Thanks for the talks!

Do we know anything about the orderability of the branched covers of these 1-bridge braid closures? (A conjecture of Allison Moore says that the branched double cover of a hyperbolic L-space knot is not an L-space.)