Components of Flip Graph of Triangulated S^3

Abstract: Let (\mathcal{F}(n)) be the graph of (n)-vertex triangulations of the 3-sphere (S^3), with edges as bistellar 2--3 and 3--2 moves. Pachner's theorem \cite{P91} shows the flip graph is connected with 1--4 and 4--1 moves, but (\mathcal{F}(n)) loses connectivity: it is connected for (5 \leq n \leq 9) ((n=5) minimal for (S^3)) but splits into multiple components at (n=16), (n=20), (n=21), and likely beyond. The polytopal closure of (\mathcal{F}(n)) is the component with all boundary complexes of convex 4-polytopes. We prove (\mathcal{F}(10)) and (\mathcal{F}(11)) are connected by showing: every non-polytopal 10-vertex seed triangulation (no 3--2 flips) is one 2--3 flip from a convex-polytope boundary, and every 11-vertex seed triangulation arises from a 10-vertex convex polytope via a 1--4 flip and 2--3 or 3--2 flips, both in the polytopal closure. We address four unflippable (S^3) complexes ((U(16)), (U(20)), (U_1(21)), (U_2(21))), showing each connects to the polytopal closure of (\mathcal{F}(n+1)) after one 1--4 vertex insertion and an annealing process maximizing removable-vertex chains. We propose the Weeping Willow Conjecture: non-polytopal components of (\mathcal{F}(n)) stem from the polytopal closure of (\mathcal{F}(m)), (m > n), via 4--1 moves, with the polytopal closure as the trunk and other components as branches.