9 problems

A critical surface is a Heegaard surface with the criticality property defined in the paper, and an index 2 minimal surface is a minimal surface whose second variation has index 2.…

Let $M$ be a 3-manifold containing unstabilized Heegaard surfaces $F$ and $G$, neither of which has topological index $1$. Suppose that the minimal-genus common stabilization of…

Let $K$ be a knot and let $F$ be a Heegaard surface such that the distance of $K$ with respect to $F$ is at least $2$. Distance conjecture. The distance of $K$ with respect to any…

Let $\Sigma_g$ be a genus-$g$ Heegaard surface in $S^3$, with $g\geq 4$, and let $\mathcal{R}(\Sigma_g)$ denote its reducing sphere complex. Its vertices are isotopy classes of red…

Let a topologically minimal surface in $S^3$ mean a surface whose disk complex is either contractible or has nontrivial homotopy groups. Bachman's conjecture. There are no topologi…

Let $S$ be a genus $n$ Heegaard surface of $S^3$, and let its topological index be the smallest integer $k$ such that $\pi_{k-1}(\mathcal{D}(S))$ is non-trivial, where…

Let $F$ be a surface of positive genus, and let $I$ denote the closed interval. A surface is topologically minimal when its disk complex is empty or non-contractible. Product-regio…

Let $M$ be either the 3-sphere $S^3$ or the 3-ball $B^3$, and let a topologically minimal surface mean a surface whose disk complex is empty or non-contractible. Topological minima…

Let $N$ be a simple 3-manifold with boundary component of genus two, and let $a$ and $b$ be basic curves on that boundary. Write $N[a]$ and $N[b]$ for the manifolds obtained by att…