The Trunk of the Restricted Flip Graph of Triangulated S^3

Abstract

Let $\mathcal{F}_M(n)$ be the restricted flip graph of $n$-vertex triangulations of a closed connected $3$-manifold $M$, whose edges are vertex-preserving $2$--$3$ and $3$--$2$ bistellar flips. Unlike the full Pachner graph, which allows vertex-changing $1$--$4$ and $4$--$1$ moves, the restricted flip graph can fragment into multiple components. We prove a general Component Preservation Theorem: for any such $M$, $1$--$4$ stellar subdivision induces a well-defined map on the connected components of $\mathcal{F}_M(n)$. For \(S^3\), we define the trunk to be the set of triangulations reachable from \(\partial\Delta^4\) using \(1\)--\(4\), \(2\)--\(3\), and \(3\)--\(2\) moves, but no \(4\)--\(1\) moves. For every \(n\ge 5\), we prove that the level-\(n\) slice of the trunk is exactly one connected component of \(\mathcal F(n)\), and that the trunk is closed upward under \(1\)--\(4\) moves. Thus any Pachner path that starts in the trunk and leaves it must do so via a \(4\)--\(1\) move. We complement these structural theorems with computational results for $S^3$. We prove that $\mathcal{F}(10)$ and $\mathcal{F}(11)$ are entirely contained within the trunk (and are therefore connected), and that all $12$-vertex seed triangulations with minimum edge valence at least $4$ lie in the trunk. Finally, we provide explicit certificates demonstrating that the four currently known isolated ``unflippable'' spheres -- $U(16)$, $U(20)$, $U_1(21)$, and $U_2(21)$ -- all enter the trunk after a single $1$--$4$ subdivision.