156 problems

Let $\mathsf{PF}_n$ denote the parking function polytope. A triangulation of a lattice polytope is regular if it is induced by a lifting function, and unimodular if every simplex i…

Permutation-degree conjecture. The labeling $L$ induces a single-valued labeling $\ell:\operatorname{Vert}(T)\to [n]$ such that

Let $(T_n(k))_{k \geq 0}$ denote the flip walk on triangulations of the sphere with $n$ vertices, and let its mixing time be measured with respect to the uniform distribution on su…

Hougardy–Lutz–Zelke conjecture. Every triangulation of an orientable surface of genus $g$ with

Let $H_g$ be a genus-$g$ handlebody, and let $\tau$ be a one-vertex triangulation of its boundary. A $[\tau]$-layered-triangulation is a layered-triangulation of $H_g$ extending…

Let $M$ be a lens space, and call a triangulation minimal if it has the smallest number of tetrahedra among triangulations of $M$. A minimal layered-triangulation of $M$ is a layer…

Let $p/q$ be a reduced rational number with $0\leq p\leq q$, including the exceptional forms $0/1$ and $1/1$. A $p/q$-layered-triangulation is a layered-triangulation of the solid…

Let M=(S^3\text{timeshspace{-1.62ex}hspace{-.4ex}hspace{.7ex}}S^1)\#({\mathbb C}{\bf P}^{\,2})^{\# 5}, and let $\text{}^4\hspace{.3pt}15^{\,4}_{\,1}$ denote the specified combi…

Good geometric triangulation conjecture. Every hyperbolic $N$ with non-empty geodesic boundary admits a good geometric triangulation.

Let an $m$-prism be triangulated using $t$ corner tetrahedra. Corner-tetrahedra conjecture. Such a triangulation has at least … tetrahedra, and therefore at least … interior triang…

Geometric realization conjecture. The triangulation $T$ is realized by hyperbolic partially truncated tetrahedra.

Let $M\in[6,\infty]$, and let $T^M_j$ be chosen randomly uniformly among isomorphism classes of spherical triangulations with $j$ vertices and maximum degree at most $M$. Given…

Ideal-prime conjecture. If the $6$-net $\sigma$ is prime, its ideal neoplatonic realization $\Delta(\sigma,\infty)$ is convex.

Ideal neoplatonic conjecture. Every $6$-net $\sigma$ has a realization $\Delta(\sigma,\infty)$, unique up to isometry, as an ideal neoplatonic.

Neoplatonic conjecture. Any $6$-net $\sigma$ has a realization, unique up to isometry, as an undented Euclidean polyhedron $\Delta(\sigma)$ built from equilateral triangles of side…

The positive spun triangulation conjecture for Vol3. For any embedded closed geodesic $\gamma$, the pair $(N,\gamma)$ does not admit a positive spun triangulation.

A double wheel is the planar triangulation obtained by joining two vertices to every vertex of a cycle. A planar triangulation is 4-connected if it has no separating set of at most…

Let $M$ be a dyadic matroid, meaning a matroid representable over the dyadic partial field. Let $P(M)$ be its matroid base polytope. Dyadic matroid triangulation conjecture. The ma…

Let $P$ be the base polytope of a polymatroid. A triangulation of $P$ is unimodular if all its simplices are unimodular with respect to the relevant lattice, and it is flag if ever…

Symmetric capacity conjecture. The symmetric capacity satisfies

Let $\mathcal{T}$ be any locally finite triangulation of $\mathbb{R}^2$ and let $f$ be a continuous piecewise linear $\varepsilon$-approximation of $F(x,y)=xy$ with respect to…

Let $\mathcal{P}_{6,48}$ be the centrally symmetric simplicial $6$-polytope whose antipodal quotient of the boundary gives a triangulation of $\mathbb{R}P^5$. A triangulation is ve…

Nullity, determinant, and FAMEDness conjecture. The following assertions hold:

Let $G$ be a 2-connected graph and let $ij$ be any edge of $G$. Let $\Gamma$ be a unimodular triangulation of the subcomplex of $P_{G\setminus ij}$ consisting of all facets visible…

A separating non-contractible cycle (SNCC) is a cycle in a triangulation that is separating and non-contractible. Let $t$ be a triangulation and let $C$ be an SNCC of $t$ of shorte…