156 problems

Seifert-fibre-edge conjecture. Every one-vertex triangulation of a small Seifert fibre space has an edge isotopic to a Seifert fibre.

Let $\mathbb{F}$ be a field. A simplicial complex $X$ is $\mathbb{F}$-tight if it is connected and, for every induced subcomplex $Y$ of $X$, the inclusion-induced map … is injectiv…

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…

Let $\sigma \\: \Gamma \to 2^V$ be a vertex-induced triangulation of a $(d-1)$-dimensional simplex, and let $k$ be a field. Generation conjecture. The local face module $L(\Gamma)$…

Complexity equality conjecture. The inequality in the bounded-complexity result can be upgraded to equality for all 42 families of knots in this paper.

Let $M$ be a hyperbolic $3$-manifold with toroidal boundary and let $X$ be an ideal triangulation of $M$. Suppose the space of angle structures $\mathscr A_X$ is non-empty. Casson…

Let $\mathbf{w} in \mathcal{V}$, and let $\mathcal{O}(Q_{\mathbf{w}})$ be the associated order polytope. A triangulation is regular if it arises from a lifting function, equivalent…

Živaljević's conjecture. Every simply connected smooth $d$-manifold admits some LC triangulations.

Let $\sigma_n$ be the average simplification path length for 3-sphere triangulations at level $n$, and let $\pi_n$ be the corresponding average for closed prime orientable 3-manifo…

Let $\mathscr{P}_1(S^3)$ be the graph whose nodes are one-vertex triangulations of the 3-sphere, arranged by level $n$, and let a simplification path be a path reducing the level t…

Vertex-minimality conjecture. For every surface $\Sigma$, the maximum excess is attained by some vertex-minimal triangulation of $\Sigma$ that contains $K_\omega$ as a subgraph. Mo…

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…