9 problems

Let $(M,\bold T)$ be a minimally triangulated closed orientable $3$-manifold. A cluster of three 2-quad-type solutions on $\bold T$ is a triple of 2-quad-type solutions whose non-z…

Tautness and minimality conjecture. These three normal surfaces are taut, and consequently the triangulations $\mathcal{U}_{k,n}$ are minimal.

Let $M$ be as in Theorem, let $\mathcal{M}\mathcal{L}_0(M)$ be the space of measured laminations without boundary in $M$, and let $\mathcal{P}_{\mathcal T}$ be the relevant polyhed…

Equality conjecture. For every normal complex surface singularity,

A triangulation $\mathcal{T}$ is a finite triangulation of a closed 3-manifold, and $S^3$ recognition asks whether $\mathcal{T}$ triangulates the 3-sphere. The complexity class…

Quadrilateral bound. For every such normal surface,

For each $n$, let $\overline{\sigma}_n$ denote the average number of vertex normal surfaces over all closed 3-manifold triangulations of size $n$, counted up to isomorphism. A vert…

Let $n$ be a positive integer other than $1,2,3,5$, and let $k$ be the integer determined by the congruence class of $n$ modulo $4$. A vertex normal surface is a vertex solution in…

Let $(M,\bold T)$ be a closed irreducible orientable minimally triangulated $3$-manifold. A solution to Haken's normal surface equation is of 2-quad-type if it has exactly one or t…