Problems in Floer homology

Problem: Classify open books with page the 4-holed sphere which correspond to tight contact structures. Recall that an open book is a pair $(S,h)$ where $S$ is a compact, oriented surface with boundary, and $h$ is a diffeomorphism of $S$ which restricts to the identity on $\partial S$. We will refer to $h$ as the monodromy  of the open book and $S$ as the page. A construction of Thurston-Winklenkemper assigns a contact 3-manifold $(M_{S,h},\xi_{S,h})$ to this open book, and Giroux proved that this assignment defines a bijective correspondence between open books up to positive stabilization and contact 3-manifolds up to contactomorphism (see here and here for recent proofs of Giroux's result). This problem is concerned with understanding how topological features of the monodromy are reflected in geometric features of the contact structure in Giroux's correspondence. For instance, Giroux proved that a contact 3-manifold is Stein fillable if and only if there is a correspondin...

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...

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...

Problem: Prove that nearly-fibered knots which aren't cables are persistently foliar. A knot $K\subset S^3$ is said to be persistently foliar if for every rational $r$ the complement $S^3\setminus n(K)$ admits a coorientable taut foliation (CTF) which meets the boundary torus $\partial n(K)$ transversely in a foliation by curves of slope $r$. Note that if a knot $K$ is persistently foliar than every nontrivial surgery on $K$ admits a CTF. The L-space and cabling conjectures together predict that if $K$ is neither a cable nor an L-space knot then every nontrivial surgery on $K$ admits a CTF. It is therefore reasonable to conjecture that such knots are also persistently foliar. This is what Delman-Roberts call the L-space knot conjecture . In particular, L-space knots are fibered. So one expects that every non-fibered, non-cable knot is persistently foliar. Li-Roberts proved that for any knot there is an interval around 0 such that for every slope in the interval there is a CTF ...

The purpose of this site is to record forward-looking problems of interest around Floer homology as used in low-dimensional topology. The focus of this list is meant to be narrower than that of the Kirby list, the problems more approachable.

Question: Does Khovanov homology exhibit representation stability with respect to cabling? Given a knot $K$ in the 3-sphere, we denote by $$\mathit{Kh}^{i,j}(K)$$ its Khovanov homology in homological grading $i$ and quantum grading $j$. Let $K_{n}$ be the $n$-cable of the knot. Grigsby-Licata-Wehrli associate to each braid $\beta \in B_n$, a link cobordism in $S^3 \times I$, $$\Sigma_\beta: K_{n} \to K_{n},$$ and prove that the induced map on Khovanov homology depends only on the permutation associated with $\beta$. In this way, each element $\sigma \in S_n$ gives rise to an automorphism $$\Phi_\sigma: \mathit{Kh}^{i,j}(K_n) \to \mathit{Kh}^{i,j}(K_n),$$ which turns the Khovanov homology of the $n$-cable in each bigrading into an $S_n$-representation. The dimensions of the vector spaces $\mathit{Kh}^{i,j}(K_n)$ are unbounded as $n \to \infty$. But the question is asking whether the decomposition of $\mathit{Kh}^{i,j}(K_n)$ into irreducible $S_n$-representations stabilizes as $n \...

(by Tye Lidman) Here are two problems pertaining to the famous L-space conjecture. The L-space conjecture predicts that if $Y$ is a prime rational homology sphere, then $Y$ is not an L-space if and only if $\pi_1(Y)$ is left-orderable if and only if $Y$ admits a co-orientable taut foliation. Ozsvath-Szabo  show that if $Y$ admits a co-orientable taut foliation then $Y$ is not an L-space, using topological results of Eliashberg/Etnyre and Eliashberg-Thurston which turn a taut foliation on $Y$ into a symplectic filling of $Y$ with $b^+ > 0$. Therefore, the L-space conjecture predicts that if $Y$ is not an L-space, then $Y$ should admit a symplectic filling of $Y$ with $b^+ > 0$. Currently, it is not clear how to show that non-L-spaces always admit symplectic fillings, even without the assumption on $b^+$. Problem: Prove that non-L-space admit symplectic fillings. This problem seems very hard in general, but one can restrict to specific families of non-L-spac...

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...