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Components of Flip Graph of Triangulated S^3

V. Faber, M. Murphy

TL;DR

This work analyzes the restricted flip graphs $\mathcal{F}(n)$ of $n$-vertex triangulations of the 3-sphere, where edges correspond to $2$--$3$ and $3$--$2$ moves. It shows $\mathcal{F}(10)$ and $\mathcal{F}(11)$ are connected by leveraging seed triangulations, polytopal closures, and a lemma ensuring $1$--$4$ flips preserve polytopality when increasing vertex count. It further demonstrates that all known outside-polytopal components become part of the polytopal closure after a single vertex insertion, via a tailored simulated annealing strategy oriented toward stacked 4-polytopes. These results motivate the Weeping Willow Conjecture, positing the polytopal closure as the graph’s trunk with non-polytopal components descending through $4$--$1$ flips, and suggest potential implications for the complexity of 3-sphere recognition and broader understanding of triangulation connectivity.

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.

Components of Flip Graph of Triangulated S^3

TL;DR

This work analyzes the restricted flip graphs of -vertex triangulations of the 3-sphere, where edges correspond to -- and -- moves. It shows and are connected by leveraging seed triangulations, polytopal closures, and a lemma ensuring -- flips preserve polytopality when increasing vertex count. It further demonstrates that all known outside-polytopal components become part of the polytopal closure after a single vertex insertion, via a tailored simulated annealing strategy oriented toward stacked 4-polytopes. These results motivate the Weeping Willow Conjecture, positing the polytopal closure as the graph’s trunk with non-polytopal components descending through -- flips, and suggest potential implications for the complexity of 3-sphere recognition and broader understanding of triangulation connectivity.

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.
Paper Structure (7 sections, 1 theorem, 1 equation, 1 figure)

This paper contains 7 sections, 1 theorem, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Polytopal Triangulations and Connectivity
  3. $\mathcal{F}(10)$ is Connected
  4. $\mathcal{F}(11)$ is Connected
  5. Analysis of Unflippable Complexes + 1 Vertex
  6. Weeping Willow Conjecture
  7. Conclusions and Future Directions

Key Result

Lemma 1

If $T$ is a polytopal triangulation on $n$ vertices, then a 1--4 flip into any face of $T$ results in a polytopal triangulation on $n+1$ vertices.

Figures (1)

  • Figure 1: The Weeping Willow Conjecture

Theorems & Definitions (3)

  • Definition 1
  • Lemma 1
  • Conjecture 1: Weeping Willow Conjecture