39 problems

Let $K_1,K_2\subset S^3$, write $K=K_1\#K_2$, and let $E(K_i)$ denote the exterior of $K_i$. Let $t(K_i)$ denote the tunnel number, and call a Heegaard splitting minimal genus if i…

Let $K_1$ and $K_2$ be knots in $S^3$, and write $K=K_1\#K_2$. Let $E(K)$ denote the exterior of $K$, and let $t(K)$ denote its tunnel number. A Heegaard splitting is weakly reduci…

Let $M_i$ ($i=1,2$) be irreducible 3-manifolds with connected boundary, with $\partial M_1\cong\partial M_2\cong F$. Let $\mathcal{D}_i$ be the set of essential curves in $F$ that…

Geometric presentation conjecture. The group $\langle x_1,\ldots,x_n\mid w\rangle$ is the fundamental group of an orientable 3-manifold if and only if $w$ is a geometric word.

Let $K\subset S^3$ be a knot that is not $\gamma$--primitive, and let a minimal genus Heegaard splitting of its exterior $E(K)$ be given. A boundary stabilization is the genus-incr…

Let $X$ and $Y$ each have a single incompressible boundary component of genus at least two, let $\varphi\colon\partial X\to\partial Y$ be a gluing map, and let $M=X\cup_\varphi Y$.…

Let $X$ have a single boundary component $F$ with $g(F)\geq 2$, and let $V\cup_H W$ be a minimal-genus Heegaard splitting of $X$ with $F$ contained in the compression body $V$. Let…

Let $X$ and $Y$ be 3-manifolds, each with a single incompressible torus boundary component, and let $M$ be obtained by gluing them along their boundaries. There is a complexity on…

Let $M$ be a closed orientable $3$-manifold and let $H\cup H^{\prime}$ be a Heegaard splitting of $M$. A closed curve in $\partial H$ is primitive in $H$ if it does not represent a…

Let $M$ be a closed orientable $3$-manifold and let $H\cup H^{\prime}$ be a Heegaard splitting of $M$. The algebraic girth is defined using the minimum girth of a nontrivial relato…

Large-genus Heegaard splitting conjecture. If $M$ contains irreducible Heegaard splittings of arbitrarily large genus, then $M$ is Haken.

Generalized Waldhausen conjecture. If $M$ contains infinitely many Heegaard splittings, pairwise non-isotopic, all of the same genus, then $M$ is toroidal.

Sedgwick's conjecture. If $M$ contains infinitely many irreducible Heegaard splittings that are pairwise non-isotopic, then $M$ is Haken.

Let $M_1$ and $M_2$ be closed, orientable $3$-manifolds, and let $H_i$ be a Heegaard surface in $M_i$ for $i=1,2$. The connected sum $H_1\# H_2$ is a Heegaard surface in…

Let $K$ and $K'$ be knots, let $K\#K'$ denote their connected sum, and let $t(K)$ denote the tunnel number. A knot admits a primitive meridian when its exterior has a minimal genus…

Let $M$ be a closed, orientable three-manifold. A Heegaard splitting is full when it does not have the disjoint curve property in the terminology of the source. Finiteness conjectu…

Let $M$ be a closed, orientable, atoroidal three-manifold. A Heegaard splitting is strongly irreducible if every compressing disk on one side intersects every compressing disk on t…

Morimoto's conjecture. The knots $K_1\subset S^3$, $K_2\subset S^3$, and $K_1\#K_2$ satisfy

A trisection of a closed 4-manifold is a decomposition into three 4-dimensional 1-handlebodies whose pairwise intersections are 3-dimensional handlebodies and whose triple intersec…

Let $M$ be a non-Haken $3$-manifold, and let $a3_1$ and $a3_2$ be strongly irreducible Heegaard surfaces in $M$ that are not isotopic. A lowest genus stabilization is a common stab…

Gordon's conjecture. The connected sum of two unstabilized Heegaard splittings is unstabilized.

Let $M$ be a tunnel-number-one orientable $3$-manifold with incompressible torus boundary. A tunnel of $M$ determines a distinguished-wave-determined slope on $\partial M$. The two…

Primitive/Seifert position conjecture. Any Seifert-fibered surgery on a hyperbolic tunnel-number-one knot arises from a primitive/Seifert position whose surface slope corresponds t…

Let a strongly irreducible Heegaard surface lie in a Riemannian three-manifold. Pitts–Rubinstein conjecture. It is either isotopic to a minimal surface of index at most $1$, or iso…

Let $M$ be a closed orientable manifold with a Heegaard surface $S$ that divides $M$ into handlebodies $V$ and $W$. Let $\mathrm{Sp}(S)$ be the spine graph of $S$, and let…