18 problems

Let $K$ be a knot in $S^3$ and let $R$ be a genus $g$ Seifert surface of $K$. Write $\widehat{HFK}(K,g)$ for the top Alexander-graded knot Floer homology and…

Let $\Lambda \subset (\Sigma \times \mathbb{R}, \xi_\Gamma)$ be a Legendrian knot. Let $Y(\Sigma, \Gamma) \subset \Sigma \times \mathbb{R}$ be the convex sutured contact manifold a…

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 $(N_3,\gamma)$ be the acylindrical taut sutured manifold obtained by cutting the 3-chain link complement along a 3-punctured sphere. Its volume is $5.33\ldots$. Minimal-volume…

Morse-Novikov genus conjecture. One expects

Sutured instanton–Heegaard Floer equivalence conjecture. For every balanced sutured manifold $(M,\gamma)$, there is an isomorphism

Let $(M,\gamma)$ be a balanced sutured manifold. Write $SHI$ for sutured instanton Floer homology, $SFH$ for sutured Heegaard Floer homology, $I^{\sharp}$ for framed instanton Floe…

Juhász's conjecture. The sutured manifold $(M,\gamma)$ admits a taut foliation of depth at most $2k$.

Connectivity conjecture. Any two balanced pairings of the same size $n$, where $n$ is odd, are connected.

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 $M$ be a taut sutured genus-two handlebody. A representation $\alpha:pi_1(M)\to\operatorname{GL}_1(\mathbb C)$ is certifying when the inclusion-induced maps … are isomorphisms.…

Let $M=(M,R_-,R_+,\gamma)$ be a taut sutured manifold. A representation $\alpha\colon\pi_1(M)\to\operatorname{SL}_2(\mathbb C)$ makes $M$ a homology product when the inclusion-indu…

Sutured ECH invariance conjecture. The sutured embedded contact homologies satisfy

Let $F,F',F”$ be connected Riemann surfaces and let $Z$ be a finite subset of the identified boundaries. Let $Y_1,Y_2$ be sutured manifolds with … and let $Y=Y_1\cup_{F'}Y_2$ be ob…

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…

Let $(M,b3)$ be a balanced sutured manifold. Denote its monopole, Heegaard, and instanton sutured homology groups by $aSHM(M,b3)$, $aSFH(M,b3)$, and $aSHI(M,b3)$, respectively. Sut…

Let $(M,\gamma)$ be a taut sutured manifold, meaning a sutured manifold satisfying the tautness condition, and suppose that $\chi(M)<0$. A finite-sheeted cover is a covering … with…