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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.06472 |
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| _version_ | 1866908999167770624 |
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| author | Faber, V. Murphy, M. |
| author_facet | Faber, V. Murphy, M. |
| contents | 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 \(\partialΔ^4\) 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06472 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Trunk of the Restricted Flip Graph of Triangulated S^3 Faber, V. Murphy, M. Combinatorics 57Q15, 57-08, 05-08 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 \(\partialΔ^4\) 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 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. |
| title | The Trunk of the Restricted Flip Graph of Triangulated S^3 |
| topic | Combinatorics 57Q15, 57-08, 05-08 |
| url | https://arxiv.org/abs/2505.06472 |