16 problems

Let $K_1,K_2\subset S^3$ be prime knots. Use $E(K)$ for the knot exterior, $g(E(K))$ for its Heegaard genus, and $t(K)$ for its tunnel number. A knot $K$ admits a $(g,n)$ position…

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 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…

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,…

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-…

Morimoto's nonemptiness conjecture. For every $n$, the families $A_n$, $B_n$, and $C_n$ are all nonempty.

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…

Let $M$ and $N$ be closed, connected, orientable $3$-manifolds. The open book genus $\mathrm{obg}(M)$ is the minimum genus among Heegaard splittings of $M$ induced by open book dec…

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…

Let $M$ be a compact $3$-manifold and let $T$ be a separating, incompressible, orientable surface of genus $g$ properly embedded in $M$. If $M_1$ and $M_2$ are the components of…

Redundancy conjecture. The hypotheses that $K'$ is hyperbolic in Theorem 8.1 and Corollary 1.1, and that $p/q$ is not a $(g,2w(g)-2)$-boundary slope in Corollary 1.1, are redundant…

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…

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

Genus-two conjecture. All hyperbolic manifolds with bicuspid fundamental group have Heegaard genus $2$.