Problems in Floer homology

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...