95 problems

For every smooth knot type $K$ that is Legendrian simple—meaning that any two Legendrian representatives $L_1,L_2$ of $K$ with $\operatorname{tb}(L_1)=\operatorname{tb}(L_2)$ and…

Let $(V,\xi)$ be a closed contact manifold, where $\xi$ is a cooriented contact structure, and let $\operatorname{Cont}_0(V,\xi)$ denote the identity component of its contactomorph…

Contact cosmetic surgery conjecture. Any Legendrian knot in $(Y,si)$ that is not smoothly an unknot admits no cosmetic contact surgeries.

ECH–knot Floer correspondence conjecture. There is an isomorphism

Let $L$ and $L'$ be Legendrian knots in the standard contact 3-sphere $(S^3,\xi_{st})$. Write $L(-1)$ and $L(+1)$ for the contact manifolds obtained by Legendrian $(-1)$- and…

Let $D_n$ be the circular spherical divisor with $n$ components discussed above. A rational embedding means an embedding of $D_n$ into a rational surface. Rational embeddability co…

Let $M$ be a $(2n+1)$-dimensional manifold equipped with an almost contact structure, meaning a codimension-$1$ cooriented distribution together with a conformally symplectic class…

Let $\beta$ be a positive braid, let $L_\beta$ be its braid-positive link closure, and let the corresponding Legendrian closure be represented by the front diagram $f_\beta$. A Leg…

Let $Y$ be a $3$-manifold and let $\xi$ be a contact structure on $Y$. Let ${\rm Tor}(Y,\xi)$ denote the torsion, defined as the supremum of the integers $n\geq 1$ for which there…

Let $Y$ be a $3$-manifold and let $\xi$ be a contact structure on $Y$. The torsion of $(Y,\xi)$, denoted by … is the supremum of the integers $n\geq 1$ for which there exists a con…

Let $L$ be a transverse representative of an alternating smooth link. Write $\psi(L)\in\widetilde{Kh}(L)$ for the transverse-link invariant in reduced Khovanov homology, let…

K-contact energy-minimization conjecture. The curl eigenform $\eta$ defined by a $K$-contact structure is always energy-minimizing on $\Xi_\eta$.

Higher-dimensional torus conjecture. There exists a $T^2$-invariant contact structure $\xi$ on $T^{2n+1}$ such that

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…

Nonfillability conjecture. No contact structure defined by Figure and satisfying conditions is fillable.

Distinctness conjecture. All the tight contact structures defined by Figure and satisfying the assumptions are distinct up to isotopy.

Finiteness conjecture. If $M$ is atoroidal, then $M$ carries only finitely many isotopy classes of tight contact structures.

Hydrodynamical Weinstein conjecture. Every smooth rotational Beltrami flow on a Riemannian three-manifold has a closed orbit.

Let $(\Sigma,\eta)$ be a contact Hamiltonian manifold, where $\eta$ is a contact Hamiltonian structure on $\Sigma$. A heterodimensional cycle is a pair of heteroclinic trajectories…

Let $\mathrm{minwr}_{\mathbb{R}^2}$ and $\mathrm{minwr}_{S^2}$ denote the minimum writhe of a max-tb Legendrian unknot front projection that is smoothly hard on the plane and spher…

Let $K$ be an orientable polyhedral Lagrangian in $(\mathbb{R}^4,\omega)$, let $v$ be a valence-$4$ vertex, and let $\Gamma_{v,\delta}\subset (S^3,\xi_{\mathrm{std}})$ be the Legen…

Let $K$ be a null-homologous smooth knot type in a tight contact manifold $(M,\xi)$. A Legendrian representative of $K$ has a Thurston–Bennequin invariant $\tb$. Low-Thurston–Benne…

Fix a smooth knot type $K$. A transverse representative $T$ of $K$ has a standard neighborhood, whose boundary has a slope, and standard neighborhoods are considered up to contact…

Let $T$ be a transverse knot in a contact manifold. A standard neighborhood of $T$ has a boundary slope, measured with respect to the chosen framing, and standard neighborhoods are…

Low Conjecture. The skies of causally related points in $X$ are topologically linked.