12 problems

Let $M$ be a closed, irreducible $3$-manifold and let $\mathcal F$ be a maximal embedded collection of orientable, incompressible surfaces. If $M-N(\mathcal F)$ has $n$ components,…

Morimoto's conjecture. The inequality

A knot exterior $X$ is the complement of an open tubular neighborhood of a knot, and $g(X)$ denotes its Heegaard genus. An exterior admits a primitive meridian if it has a minimal-…

Let $M=\mathcal{W}\cup_\mathcal{T}\mathcal{V}$, where $\mathcal{T}=\partial \mathcal{W}=\partial \mathcal{V}=\mathcal{W}\cap\mathcal{V}$ is a genus-$g$ surface. Let…

Let $M$ and $N$ be closed orientable 3-manifolds. A degree-one map is a map $f\colon M\to N$ of degree one, and $g(M)$ denotes the Heegaard genus of $M$. The degree-one map conject…

Rank versus genus conjecture. It holds that

For a knot $K\subset S^3$, let $r(K)$ be the minimal number of generators of $\pi_1(S^3-K)$, and let $E(K,S^3)$ denote its exterior. The rank-genus conjecture. For all knots…

Universal tunnel-number bound conjecture. There is a function

Lackenby's Heegaard gradient conjecture. If $\chi^h_{\mathcal{L}}(M)=0$ for a family of covers, then $M$ virtually fibres over a circle. The source motivates this by noting that vi…

Let $M$ be a finite-volume hyperbolic $3$-manifold and let $\{M_i\longrightarrow M\}$ be a family of finite-sheeted covers. Write $\operatorname{rgr}(M,\{M_i\})$ for the rank gradi…

Let $M$ be an orientable finite-volume hyperbolic $3$-manifold. Its rank is the minimal number of generators of $\pi_1(M)$, and its Heegaard genus is the minimal genus of a Heegaar…

Heegaard–discrete Morse conjecture. Some closed $3$-manifold with Heegaard genus $g$ admits discrete Morse functions with less than $g$ critical edges.