9 problems

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…

A discrete manifold is a graph of degree $|\pi|=n+1$ such that, for every vertex $u\in V(G)$, a sequence of graph-local Pachner moves sends $\operatorname{Star}(G,u)$ onto…

A four-dimensional Riemannian Spin Foam model with simplicity constraints is considered on triangulations related by Pachner moves, including the $5$--$1$ and $4$--$2$ moves. In th…

Extension conjecture. The analogues of these two theorems hold also for a $2\to 4$ Pachner move.

For a given 3-manifold $\mathcal{M}$, let $\phi_n(\mathcal{M})$ denote the fraction of nodes at level $n$ of $\mathscr{P}_1(\mathcal{M})$ that have an arc leading directly down to…

The Grassmann-algebraic Pachner-move invariant. The expression

Consider the $2\to4$ Pachner move replacing the 4-simplices $(12345)$ and $(12346)$ by $(12356)$, $(12456)$, $(13456)$, and $(23456)$. Let $\tilde{\mathcal W}_{ijklm}$ denote the c…

Let $M$ be a compact three-dimensional manifold with triangulated boundary, and let $\vec I(M)$ be its invariant vector. A linear transformation $\mathcal A$ is constructed from a…