20 problems

Let $(M,\Gamma,\xi)$ be a sutured contact $3$-manifold. For $A\in H_1(M;\mathbb{Z})$, let $\mathfrak{s}_\xi$ be the canonical Spin$^c$-structure determined by $\xi$, let…

Let $(Y_1,\lambda_1)$ and $(Y_2,\lambda_2)$ be closed connected contact three-manifolds with nonzero contact invariants, and let $k$ be a nonnegative integer. Write…

Let $(Y,\Lambda)$ be a pair consisting of a closed contact $3$-manifold $Y$, a closed Legendrian link $\Lambda \subset Y$, and a contact form $\alpha$ that is non-degenerate for…

Large-parameter $U$-map conjecture. For $*$ sufficiently large, the $U$ map is an isomorphism on the chain level of the ECH of prequantization bundles, for the chain complex of Pro…

ECH–knot Floer correspondence conjecture. There is an isomorphism

Sutured ECH–SFH isomorphism conjecture. There is an isomorphism

Let $(M,\Gamma,\xi=\ker\alpha)$ be a sutured contact manifold, let $A\in H_1(M;\mathbb{Z})$, and let $\mathfrak{s}_\xi$ be the relative Spin$^c$ structure determined by $\xi$. Writ…

Let $L$ be a link in a contact three-manifold $Y$, and let $\alpha$ be any contact form on $Y$ adapted to $L$. Denote by $ECK(L,Y,\alpha)$ and $\widehat{ECK}(L,Y,\alpha)$ the full…

Let $K$ be a knot in a contact three-manifold $Y$, and let $\alpha$ be any contact form on $Y$ adapted to $K$. Denote by $ECK(K,Y,\alpha)$ the full embedded contact knot homology a…

Let $X_\Omega$ be a smooth convex toric domain with boundary $Y$. Let $C_*^{T^3}$ be the chain complex freely generated over $\mathbb{Z}/2$ by torus generators, with differential g…

Naturality conjecture. The diagram of filtered Seiberg–Witten Floer cohomology groups, with horizontal maps given by the chain maps of and vertical maps given by the canonical iden…

Let $(Y,\xi)$ be a closed contact $3$-manifold. Let $(\lambda,J)$ range over ECH data for $(Y,\xi)$, and let $f(Y,\lambda,J)$ be the invariant defined in the paper. Converse overtw…

Twisted ECH cobordism conjecture. If $c(M_+,xi_+;\Omega_+)$ vanishes, then $c(M_-,xi_-;\Omega_-)$ also vanishes. Moreover, if $c(M_+,xi_+;\Omega_+)$ lies in the image of $U^k$ on…

General ECH volume conjecture.

Strict ECH volume bound.

Volume conjecture.

Sutured ECH–SFH conjecture. There should be an isomorphism

Sutured ECH invariance conjecture. The sutured embedded contact homologies satisfy

Let $(M,\Gamma,U(\Gamma),\xi)$ be a sutured contact $3$-manifold with adapted contact form $\alpha$, let $(M^{\ast},\alpha^{\ast})$ be its completion, and let $J$ be an almost comp…

Let $(M,\Gamma,\xi)$ be a contact $3$-manifold with convex or sutured boundary and dividing set $\Gamma$. Let $\mathfrak s_{\xi}$ denote the relative Spin$^c$-structure determined…