Knot Floer homology and pseudo-Anosov flows

Problem: Is the knot Floer homology of a hyperbolic knot in its next-to-top Alexander grading related to the number of orbits of the pseudo-Anosov flow on its complement which generate the first homology of the complement?

If $K\subset Y$ is a hyperbolic knot then its complement admits a pseudo-Anosov flow. This follows from an unpublished result of Gabai, a proof of which was outlined by Mosher; see this paper of Landry--Tsang for some of the history. If $K$ is fibered, then the suspension flow of its monodromy is pseudo-Anosov, and in this case, the answer to the problem is yes. Indeed, work of Ni and independently Ghiggini--Spano, which relies on (1) a relationship between periodic Floer homology and monopole Floer homology due to Lee--Taubes, and (2) an isomorphism between monopole and Heegaard Floer homology due to Kutluhan--Lee--Taubes, shows that if $K$ is fibered then $$\widehat{\mathit{HFK}}(Y,K,g-1)-1$$ is an upper bound for the number of fixed points of the pseudo-Anosov representative of its monodromy. These fixed points correspond to the homology-generating orbits of the induced pseudo-Anosov flow on the knot complement. So the question is really asking whether something analogous holds for non-fibered knots as well.

Baldwin--Vela-Vick proved that if $K\subset Y$ is fibered then $$\dim\widehat{\mathit{HFK}}(Y,K,g-1) \geq \dim\widehat{\mathit{HFK}}(Y,K,g),$$ where the latter is equal to 1 in this case. It is conjectured that this inequality holds for non-fibered knots as well. Perhaps then (and this is admittedly very speculative) it is the quantity $$\dim\widehat{\mathit{HFK}}(Y,K,g-1) -\dim\widehat{\mathit{HFK}}(Y,K,g)$$ which is related to the number of degree-1 orbits for general hyperbolic non-fibered knots. 

We mention below some more tractable problems. The first is to prove the a strengthened version of the inequality above:

Problem: Prove for any knot $K\subset Y$ that there is an injective differential in the knot Floer complex taking one of the two forms 

  • $\widehat{\mathit{HFK}}(Y,K,g) \to\widehat{\mathit{HFK}}(Y,K,g-1),$ or 
  • $\widehat{\mathit{HFK}}(Y,K,1-g) \to\widehat{\mathit{HFK}}(Y,K,-g),$ 

after omitting a small number of $Y$ such as $\#^g S^1\times S^2$.

There should be a Heegaard-diagrammatic approach to this similar to what's done for fibered knots by Baldwin--Vela-Vick (non-fibered knots have analogous Heegaard diagrams associated with the broken fibrations of their complements). The primary difficulty with that approach was in showing that the knot Floer complex in filtration level $-g$ is generated by the obvious contact generator. This was tricky because in the standard doubly-pointed Heegaard diagram for $K$ associated with its fibration there are many other generators in the bottommost filtration level. Perhaps it would be easier to use a generalization for non-fibered knots of the Heegaard diagram studied by Ozsvath--Szabo in their original paper on the contact invariant, which has just one generator in the bottommost filtration level for a fibered knot.

Ni used the existence of such a differential for fibered knots to show that if $K$ is fibered and its monodromy is neither right-veering nor left-veering then $$\dim\widehat{\mathit{HFK}}(Y,K,g-1) \geq 2.$$ The monodromy being veering (right or left), at least for hyperbolic knots, is equivalent by Honda--Kazez--Matic to its fractional Dehn twist coefficient being nonzero, which is the same as saying that the degeneracy slope of the suspension of the stable foliation of the pseudo-Anosov monodromy is not a meridian of $K$. One can talk about the degeneracy slope for a non-fibered knot as well, associated with the stable lamination of the pseudo-Anosov flow on its complement. The problem below asks whether there is an analogue of Ni's result for non-fibered knots:

Problem: Is there a relationship, for any hyperbolic knot $K\subset Y$, between $$\dim\widehat{\mathit{HFK}}(Y,K,g-1)$$ and whether the degeneracy slope of $K$ is a meridian.

For instance, could it be that degeneracy slope zero implies that $\dim\widehat{\mathit{HFK}}(Y,K,g-1) > \dim\widehat{\mathit{HFK}}(Y,K,g)?$

For the last problem below, recall that Baldwin--Ni--Sivek give a knot Floer theoretic criterion for a fibered knot which completely determines whether its monodromy is right-veering, where the latter is equivalent to the degeneracy slope of the knot being positive. The following problem asks whether there an analogue of their result for non-fibered knots:

Problem: Is it the case for arbitrary hyperbolic knots $K\subset Y$ that the degeneracy slope of $K$ is positive if and only if there is no differential in the knot Floer complex $$\widehat{\mathit{HFK}}(Y,K,g) \to\widehat{\mathit{HFK}}(Y,K,g-1),$$ after perhaps omitting a small number of $Y$ such as $\#^g S^1\times S^2$?

Baldwin--Ni--Sivek proved their result using the link between the knot Floer homology of a fibered knot and the fixed point dynamics of its monodromy, so perhaps a solution to the first problem would be helpful in solving the last.

Finally, we remark that another reason to be interested in these problems is that their solutions may enable one to prove that the 0-trace detects a host of non-fibered knots, akin to what Baldwin--Sivek prove about L-space knots here; see Section 1.2 of that paper.


Comments

Post a Comment

Popular posts from this blog

L-spaces and F-summands

Problem:  Prove that if $S^3_r(K)$ is not an L-space then its Heegaard Floer homology has a direct $\mathbb{F}$-summand. The L-space conjecture predicts that if an irreducible closed 3-manifold $Y$ is not an L-space then $Y$ admits a coorientable taut foliation. Recently, Lin proved in a short note that if $Y$ admits a coorientable taut foliation then its Heegaard Floer homology $\mathit{HF}^+(Y)$ has a direct $\mathbb{F}$-summand.  For background: recall that $\mathit{HF}^+(Y)$ is a module over $\mathbb{F}[U]$ where $\mathbb{F} = \mathbb{Z}/2\mathbb{Z}$. A direct $\mathbb{F}$-summand of $\mathit{HF}^+(Y)$ refers to a direct summand isomorphic as an $\mathbb{F}[U]$-module to $\mathbb{F}[U]/U$. Lin's result therefore gives a potential way to disprove the L-space conjecture: find an irreducible non-L-space whose Heegaard Floer homology does not have a direct $\mathbb{F}$-summand. Alternatively, one could try to prove that Lin's result cannot possibly give a counterexample by sh...
Read more

Knot Floer homology and twist coefficients

Question: What information does the knot Floer complex of a fibered knot record about the fractional Dehn twist coefficient of its monodromy? Let $\phi$ be a diffeomorphism of a surface $S$ with connected boundary, which restricts to the identity on the boundary. The fractional Dehn twist coefficient of $\phi$, denoted by $c(\phi)$, is a measure of the twisting near the boundary of the surface in the free isotopy between $\phi$ and its Nielsen--Thurston representative. It was defined by Honda-Kazez-Matic as a quantification of their notion of right-veering surface diffeomorphisms, which in turn came from their study of contact structures on 3-manifolds (a contact structure is tight if and only if all of its supporting open books have right-veering monodromy). It is also closely related to and inspired by Gabai's notion of degeneracy slope . Suppose for example that $\phi$ is freely-isotopic to a pseudo-Anosov map $\phi_0$. The latter fixes a pair of transverse, singular, tran...
Read more

Left-orderability and SU(2)-representations

Problem: Prove that if $Y$ is an irreducible 3-manifold with left-orderable fundamental group then there exists a homomorphism $\pi_1(Y)\to SU(2)$ with nonabelian image. Suppose $Y$ is irreducible with left-orderable fundamental group. Then the L-space conjecture predicts that $Y$ is not an L-space. Together with Kronheimer-Mrowka's conjectured isomorphism $$\widehat{\mathit{HF}}(Y;\mathbb{C}) \cong_{\mathrm{Conj.}} I^\sharp(Y;\mathbb{C})$$ between Heegaard Floer homology and framed instanton homology, this would imply that $Y$ is not an instanton L-space. If $Y$ satisfies a further mild non-degeneracy condition (and perhaps even if $Y$ doesn't) then being an instanton non-L-space implies that there exists a homomorphism $\pi_1(Y)\to SU(2)$ with nonabelian image, per  Baldwin-Sivek . While these conjectures serve as motivation, one can try to prove the stated link between left-orderability and nonabelian $SU(2)$-representations directly. That's the goal of this problem...
Read more