295 problems

Aldous's conjecture. The relaxation time of the triangulation flip walk should be $\Theta(n^{3/2})$. This is a classical mixing-time conjecture for a Catalan structure; the supplie…

Minimality conjecture. For $p,q2$, the triangulations $\mathcal{T}_{K(p,q)}$ realise the triangulation complexity of the complement of $K(p,q)$; equivalently, they use the minimal…

Kühnel–Lutz conjecture. The triangulation of an $n$-manifold is tight if and only if it is strongly minimal.

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…

Let $N$ be a compact $3$-manifold with boundary, and let $\mathcal{T}_N$ be an ideal triangulation of $N$. A truncated hyperbolic tetrahedron realization assigns hyperbolic truncat…

Every compact hyperbolic $3$-manifold with totally geodesic boundary is a compact $3$-manifold equipped with a complete hyperbolic metric whose boundary is totally geodesic. A geom…

Let $M$ be a hyperbolic $3$-manifold with toroidal boundary, let $X$ be an ideal triangulation of $M$, and let $\gamma\in\pi_1(\partial M)$ be a simple closed curve. For…

Geometric triangulation conjecture. Every cusped hyperbolic $3$-manifold admits a geometric triangulation.

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

Let $M$ be a matroid and let $P_M$ denote its base polytope. A triangulation of $P_M$ is unimodular if every simplex in the triangulation is unimodular with respect to the lattice…

Let $\sigma \\: \Gamma \to 2^V$ be a vertex-induced triangulation of a simplex, and let $k$ be a field of characteristic $0$. A generic special linear system of parameters is the s…

Dihedral symmetry conjecture. The well-known lattice triangulation of $\Delta_{k,n}$ is invariant under the dihedral group $\langle a,b\rangle=D_{2n}\leq S_n$, and $D_{2n}$ is the…

Triangulation Conjecture. Any topological manifold admits a triangulation.

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 $L(\Gamma)$…

Matveev–Jaco–Rubinstein conjecture. For $n>3$,

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…

Let $K$ be one of the complexes described by Brehm and Kühnel: a triangulation of an $8$-manifold different from the sphere on $15$ vertices. Brehm–Kühnel's conjecture. These compl…

Let $\Delta$ be a triangulation of a simply connected polygonal domain in $\mathbb{R}^2$, and let $C^r_k(\Delta)$ be the vector space of splines of smoothness $r$ and degree at mos…

Lutz's conjecture. The only manifolds with $\beta_1(M,\mathbb{F})=3$ admitting a $13$-vertex triangulation are $(S^2\times S^1)^{\# 3}$ and $(S^2\uplus S^1)^{\# 3}$.

Björner–Lutz conjecture. The triangulation poincare is the smallest triangulation of $\Sigma^3$ and is the unique triangulation of $\Sigma^3$ with this $f$-vector.

A $5/6^*$-triangulation of a $3$-manifold is a triangulation in which every edge has degree $5$ or $6$, meaning that $5$ or $6$ tetrahedra meet around each edge, and every triangle…

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…

Stiefel's conjecture. The class $w_{n-i}(M)$ can be defined simply as the sum of all $i$-simplices in the barycentric subdivision of a triangulation of $M$.