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 ...