36 problems

ECH–knot Floer correspondence conjecture. There is an isomorphism

Let $(Y,\gamma)$ be a sutured manifold, and let $ECH_{\mathrm{sut}}(Y,\gamma)$ and $SFH(Y,\gamma)$ denote its sutured embedded contact homology and sutured Heegaard Floer homology,…

Let $K$ be a transverse knot in a contact three-manifold $(Y,\alpha)$. Denote by $\widehat{ECK}(K,Y,\alpha)$ the hat version of embedded contact knot homology, and by…

ECH contact invariant conjecture. The homology class $c(\lambda)$ depends only on the contact structure $\xi$.

Embedded contact homology invariance conjecture. One has $\partial^2=0$; $\widetilde{ECH}_*(Y,\lambda;\Gamma,G)$ is independent of $J$; and…

ECH–Floer correspondence conjecture. Up to a grading shift,

Let $X\subset\mathbb{R}^4$ be a generic star-shaped domain with smooth boundary. Its ECH capacities satisfy … where $e_k(X)$ is the error term, and let $\operatorname{Ru}(X)$ denot…

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 $b\subset S^3$ be a transverse knot, let $\operatorname{rot}(b)$ denote its rotation number, and let $c_k^{\operatorname{link}}(b,\operatorname{rot}(b))$ denote the associated…

Let $(Y,\lambda)$ be a closed contact $3$-manifold and let $b$ be a null-homologous transverse knot. Write $\widehat{ECK}(b,Y,\lambda)$ for the hat version of knot embedded contact…

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…

Let $(Y,\omega_A)$ be a rational projective surface with an ample $\mathbb{R}$-divisor $A$ such that … and suppose $Y\setminus\operatorname{supp}(A)$ is deformation-equivalent to a…

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…

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…

Hutchings's formula conjecture. Hutchings's formula should remain true in most, and probably all, cases where $\mu(U)$ touches one or both axes.

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.