Problems in Floer homology

Let $(\Sigma,\alpha,\beta,\gamma)$ be a trisection diagram for a closed 4-manifold $X$, in the sense of Gay--Kirby . After fixing a basepoint and the usual symplectic and almost complex data, one can define an $A_\infty$-algebra with underlying chain complex given by \[\mathcal{A} = \bigoplus_{i,j\in \{\alpha,\beta,\gamma\}} \widehat{\mathit{CF}}\,(i,j),\] where the multiplication maps count pseudo-holomorphic polygons in the standard way. In other words, $\mathcal{A}$ is the subcategory of the Fukaya category of $\mathrm{Sym^g}(\Sigma\setminus \{z\})$ generated by the Lagrangian tori $\mathbb{T}_\alpha, \mathbb{T}_\beta,\mathbb{T}_\gamma.$ Up to quasi-isomorphism of $A_\infty$-algebras, this should be invariant under isotopies and handleslides, and under changes to the other auxiliary data. In other words, the quasi-isomorphism type of $\mathcal{A}$ should be an invariant of the underlying trisection of $X$. The question below asks whether this is an interesting invariant: Question:...

Question:  Does $\textrm{dim}_\mathbb{F}\widetilde{\mathit{Kh}}(L) =\textrm{dim}_\mathbb{F} \widehat{\mathit{HF}}(\Sigma(L))$ imply that $\textrm{dim}_\mathbb{F}\widetilde{\mathit{Kh}}(L)=\textrm{det}(L)$? Ozsvath--Szabo prove for any link $L\subset S^3$ that there exists a spectral sequence from the reduced Khovanov homology $\widetilde{\mathit{Kh}}(L)$ to the Heegaard Floer homology $\widehat{\mathit{HF}}(\Sigma(L))$ of the branched double cover of $S^3$ branched along $L$, with coefficients in $\mathbb{F}=\mathbb{Z}/2\mathbb{Z}$. This implies an inequality of dimensions \[\textrm{dim}_\mathbb{F}\widetilde{\mathit{Kh}}(L) \geq \textrm{dim}_\mathbb{F} \widehat{\mathit{HF}}(\Sigma(L)).\] This is an equality ---equivalently, the spectral sequence collapses---whenever \[\textrm{dim}_\mathbb{F}\widetilde{\mathit{Kh}}(L)=\textrm{det}(L)\] (which happens for instance whenever $L$ is quasi-alternating or more generally Khovanov-thin ) ,  in which case $\Sigma(L)$ is an L-...

Problem: Prove that none of the Conway knots are slice. Piccirillo is well-known for having shown that the Conway knot is not slice. The Conway knot  $C$ is a certain Conway mutant of the Kinoshita-Terasaka knot $KT$. The latter is a symmetric union and hence slice. Many knot invariants have trouble telling a knot apart from its mutant. For example, we know thanks to Kotelskiy--Watson--Zibrowious that Rasmussen's $s$ invariant doesn't see mutation. Therefore \[s(C) = s(KT) = 0,\] providing no information about the sliceness of $C$. The Alexander polynomial is also insensitive to mutation, so \[\Delta_C(t) = \Delta_{KT}(t) = 1,\] which implies that $C$ is topologically slice by Freedman. These explain why the problem Piccirillo solved was tricky. Perhaps less well known is that $C$ is merely the first member of a bi-infinite family of Conway knots. For integers $r\geq 2$ and $n\geq 1$, Kinoshita and Terasaka studied the knots $KT_{r,n}$ shown below (the boxes containing numb...

Suppose that $(S,\varphi)$ is an open book with connected binding for a contact 3-manifold $(Y,\xi)$, and suppose the fractional Dehn twist coefficient of $\varphi$ is bigger than 1. Roberts proved in this case that $Y$ admits a co-orientable taut foliation which is transverse to the binding. Eliashberg--Thurston's construction (together with its improvements by Bowden and Roberts--Kazez ) therefore produces tight contact structures $\xi_\pm$ on $\pm Y$ which are weakly semi-filled by a symplectic form on $Y\times [0,1]$. Honda--Kazez--Matic proved moreover that $\xi = \xi_+$ for one of these Roberts-type foliations.  Problem: Find a way of producing from $(S,\varphi)$ an open book supporting the other contact structure $(-Y,\xi_-)$.   One has to be a bit careful about how to interpret this because it's not a priori true that $\xi_-$ is even an invariant of the foliation (two negative perturbations of the foliation may not be isotopic). Note that $(S,\varphi^{-1})$ is ...

Let $Y$ be a closed 3-manifold. Let $S_Y$ be the set of compact, oriented, connected surfaces with boundary in $Y$. Let $\sim$ denote the equivalence relation generated by isotopy and positive Hopf plumbing, and consider the set $$\Sigma_Y= S_Y/\sim$$ of equivalence classes under this relation. Problem: is there a natural geometric description of the set $\Sigma_Y$? Let $S_Y^f\subset S_Y$ be the subset whose elements are fiber surfaces for fibered links in $Y$, and let $\Sigma_Y^f$ be the corresponding set of equivalence classes. Then the elements of $\Sigma_Y^f$ are in bijection with isotopy classes of contact structures on $Y$. The problem is asking about the geometric content of equivalence classes of non-fiber surfaces.  Problem: define a Floer-theoretic invariant of elements in $\Sigma_Y$. The Ozsvath-Szabo contact invariant assigns to each $\xi \in \Sigma_Y^f$ (aka each isotopy class of contact structures on $Y$) an element  $$c(\xi)\in \widehat{\mathit{HF}}(-Y).$$ Perh...

Problem:  How is the knot Floer homology of a hyperbolic knot in its next-to-top Alexander grading related to the orbits of pseudo-Anosov flows 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 problem has been answered . 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 repr...

Problem: Categorify the Fox-Milnor criterion for slice fibered knots. If $K\subset S^3$ is a slice knot (either topologically or smoothly) then its Alexander polynomial satisfies $$\Delta_K(t) = f(t)f(t^{-1})$$ for some $f\in \mathbb{Z}[t,t^{-1}]$. This is called the Fox-Milnor criterion . Knot Floer homology is said to categorify  the Alexander polynomial; more precisely, its graded Euler characteristic satisfies $$\chi(\widehat{\mathit{HFK}}(K)) = \bigoplus_{m,a}(-1)^m t^a \dim \widehat{\mathit{HFK}}(K)  = \Delta_K(t).$$ It is therefore natural to ask (the admittedly vague question of) whether there are extra features of the knot Floer homology of slice knots which recover the Fox-Milnor criterion for the Alexander polynomial. One would hope moreover that these features provide a stronger obstruction to slicing a knot than the classical Fox-Milnor criterion. This is one thing that could be meant by the charge to categorify  Fox-Milnor.  It's not at all clear howev...

Annular Khovanov homology assigns to a link $L$ in a solid torus (aka a thickened annulus) a tri-graded abelian group $$\mathit{AKh}(L) = \bigoplus_{i,j,k} \mathit{AKh}^{i,j}(L,k).$$ Grigsby-Licata-Wehrli proved that there's an $\mathfrak{sl}_2(\mathbb{C})$ action on annular Khovanov homology with complex coefficients in which the weight space grading corresponds to the $k$-grading on $\mathit{AKh}.$ It follows that annular Khovanov homology is trapezoidal with respect to the $k$-grading. More precisely, if we let $a_k = \dim\mathit{AKh}(L,k)$, then $$a_k \geq a_{k'}$$ when $k\equiv k'\,\mathrm{mod}\, 2$ and $|k|<|k'|$. In particular,  the sequence $$\{a_k\}_{k\equiv i\, \mathrm{mod}\, 2}$$ is unimodal for any fixed $i$. The question below asks whether something stronger is true: Question: For a fixed $i$, is the sequence $\{a_k\}_{k\equiv i\, \mathrm{mod}\, 2}$ in fact log-concave ? This is asking whether $$a_{k}^2 \geq a_{k-2}a_{k+2}$$ for all $k$. If so, then th...

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...