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

TL;DR

This paper studies the structure of restricted flip graphs of triangulations of the 3-sphere, proving connectivity properties of the 'trunk' subset and computationally analyzing specific cases and unflippable spheres.

Contribution

It introduces the concept of the trunk in restricted flip graphs, proves its structural properties, and provides computational evidence for connectivity and unflippable spheres in the case of the 3-sphere.

Findings

The level-n slice of the trunk forms a single connected component for all n≥5.

The entire graphs for 10 and 11 vertices are contained within the trunk, hence connected.

Known unflippable spheres become flippable after a single subdivision.

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.