12 problems

Let $M$ be a closed orientable irreducible atoroidal 3-manifold with positive first Betti number. An integral class is a class $a\in H^2(M;\mathbb{R})$ arising from integral cohomo…

L-space conjecture. The following statements are equivalent:

Yazdi's virtual Euler class one conjecture. For any even integral lattice point $w\in H^2(M;{\mathbb R})$ of dual Thurston norm $1$, there exists some finite cover $\tilde{M}$ of…

Let $K$ be a knot in $S^3$ with Seifert genus $g>0$. Let $\mathcal{S}_K\subset\mathbb{Q}$ be the set of rational slopes $s$ for which there exists a taut $\mathcal{C}^0$-foliation…

L-space conjecture. The following statements are equivalent for $M$: (1) $M$ is a non-L-space. (2) $\pi_1(M)$ is left orderable. (3) $M$ admits a co-orientable taut foliation…

Let $K\subset S^3$ be a knot, and let $S^3(K)$ denote its knot complement. Consider the irreducible homomorphisms … that map a chosen meridian $m$ to $\mathbf{i}\in SU(2)$. Assume…

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

A composite knot is a nontrivial connected sum of knots, and a knot is persistently foliar if, for each boundary slope, its complement admits a co-oriented taut foliation meeting t…

A knot $\kappa$ in $S^3$ is an L-space knot if some non-trivial Dehn surgery on $\kappa$ yields an L-space, and it is persistently foliar if, for each boundary slope, its complemen…

Let $M$ be a rational homology $3$-sphere, meaning that $H_1(M;\mathbf{Q})=0$. Its Heegaard Floer homology has rank at least $\left|H_1(M;\mathbf{Z})\right|$, and $M$ is an L-space…

Let $K$ be a knot in $S^3$ that is not an L-space knot, meaning that no nontrivial Dehn surgery on $K$ yields an L-space. A persistently foliar branched surface in the exterior of…

Let $k$ be a nontrivial knot in $S^3$, and let $M=S^3\setminus\operatorname{int}(N(k))$ be its exterior. For slopes $s\in(-a,b)$, require that $M$ have a taut foliation whose restr…