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

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 $\mathrm{\partial }{\mathrm{\Delta }}^4$ 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.