12 problems

Let $\mathcal M_k$, for $k\geq 0$, be the set of closed, oriented, smooth $3$-manifolds admitting open book decompositions whose monodromies contain at most $k$ negative Dehn twist…

Arbitrarily large Weinstein $\mathcal{L}$-invariant conjecture. The Weinstein domains in Example 1 and Example 2 have arbitrarily large Weinstein $\mathcal{L}$-invariants under thi…

The open-book conjecture.

Open-book subsurface conjecture. $(Y,\xi)$ is tight if and only if every subsurface of the page of each supporting open book is Thurston norm minimizing and, moreover, has minimal…

Planarity conjecture. No tight, positive contact structure on a Brieskorn homology sphere is supported by a planar open book decomposition.

Planar filling finiteness conjecture. Every planar contact $3$-manifold has at most finitely many distinct deformation classes of minimal symplectic fillings.

Relative Giroux correspondence. Every compact contact manifold with convex boundary is contactomorphic to

Minimum-genus alpha-Bennequin surface conjecture. If $\delta(\mathcal{T},\alpha)\geq 0$, then $\mathcal{T}$ bounds a minimum-genus $\alpha$-Bennequin surface with respect to…

Let $(Y,\xi)$ be a contact 3-manifold. The binding number $bn(\xi)$ is the minimal number of binding components among open books supporting $\xi$ whose pages have minimal genus. Th…

Let $(M,\xi)$ be a closed contact three-manifold supported by a genus one open book $(S_{1,r},\phi)$, where $\phi$ is pseudo-Anosov and the associated foliations have two singulari…

Left-handed stabilization conjecture. The left-handed stabilization of $(M,\alpha)$ along $L$ has vanishing contact homology:

Plumbing conjecture. There are deformations $\tilde\psi_i$ of $\psi_i$, isotopic to $\psi_i$ as symplectomorphisms, such that the open book