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 also allows the vertex-changing (1)--(4) and (4)--(1) moves, the restricted flip graph may have multiple connected components. We prove a general Component Preservation Theorem: for any such (M), (1)--(4) stellar subdivision induces a well-defined map from the connected components of (\mathcal{F}_M(n)) to those of (\mathcal{F}_M(n+1)). For (S^3), we define the trunk to be the set of triangulations reachable from (\partialΔ^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 results with computations for (S^3). We prove that every (10)- and (11)-vertex triangulation lies in the trunk, and hence that (\mathcal F(10)) and (\mathcal F(11)) are connected. We also prove that all (12)-vertex seed triangulations with minimum edge valence at least (4) lie in the trunk. Finally, we give explicit certificates showing that the four 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.