Unimodal connectivity by 2–3 and 3–2 Pachner moves
Let $M$ be a compact connected 3-manifold with no 2-sphere boundary components. Triangulations are generalised: finitely many tetrahedra whose faces are all identified in pairs by affine maps, with no edge identified with itself in reverse. A vertex is internal if its link is a 2-sphere and ideal otherwise; truncating the ideal vertices gives the boundary. If $M$ is closed, let $T$ and $U$ be triangulations of $M$; if $M$ has nonempty boundary, let $T$ and $U$ be ideal triangulations of $M$. Suppose that each has at least two tetrahedra and that they have the same number, possibly zero, of internal vertices. Matveev and Piergallini proved independently that any two one-vertex triangulations of a closed 3-manifold, each with at least two tetrahedra, are related by a finite sequence of 2–3 and 3–2 moves, and Benedetti and Petronio (closed triangulations with any number of vertices) and Amendola (ideal triangulations) extended this to all $T$ and $U$ as above. Burton and He proved that $T$ and $U$ are related by a sequence of 2–3 moves followed by a sequence of 2–0 moves. In their experiments on a census of minimal triangulations of closed irreducible 3-manifolds, sequences of 2–3 moves followed by 3–2 moves joined every pair of minimal triangulations of the same manifold, apart from one manifold for which the search was abandoned.
Problem. Can $T$ always be transformed into $U$ by a finite sequence consisting first only of 2–3 moves and then only of 3–2 moves, each move preserving the manifold represented? Equivalently, do $T$ and $U$ have a common triangulation that can be reached from each of them by 2–3 moves alone? Either part of the sequence may be empty.
Source: Burton and He (2026), Conjecture 3.1 (Conjecture 8 in arXiv:2012.02398v3). Stated for connected manifolds by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- GT — Geometric Topology
- Secondary
- CO
- Keywords
- 3-manifold triangulations, AIM Open Applied Problems, Pachner moves, unimodal sequences
References
@article{burton2026,
author = {Burton, Benjamin A. and He, Alexander},
title = {Connecting 3-manifold triangulations with unimodal sequences of elementary moves},
journal = {Discrete Comput. Geom.},
volume = {75},
number = {4},
year = {2026},
pages = {1306--1330},
doi = {10.1007/s00454-025-00735-4},
eprint = {2012.02398},
archivePrefix = {arXiv},
}
@article{matveev1988,
author = {Matveev, S. V.},
title = {Transformations of special spines and the {Z}eeman conjecture},
journal = {Math. USSR-Izv.},
volume = {31},
number = {2},
year = {1988},
pages = {423--434},
doi = {10.1070/IM1988v031n02ABEH001083},
}
@article{piergallini1988,
author = {Piergallini, Riccardo},
title = {Standard moves for standard polyhedra and spines},
journal = {Rend. Circ. Mat. Palermo (2) Suppl.},
number = {18},
year = {1988},
pages = {391--414},
}
@article{benedetti1995,
author = {Benedetti, Riccardo and Petronio, Carlo},
title = {A finite graphic calculus for 3-manifolds},
journal = {Manuscripta Math.},
volume = {88},
number = {1},
year = {1995},
pages = {291--310},
doi = {10.1007/BF02567824},
}
@article{amendola2005,
author = {Amendola, Gennaro},
title = {A calculus for ideal triangulations of three-manifolds with embedded arcs},
journal = {Math. Nachr.},
volume = {278},
number = {9},
year = {2005},
pages = {975--994},
doi = {10.1002/mana.200310285},
}
@misc{aim-unimodal-pachner-connectivity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Unimodal connectivity by 2–3 and 3–2 Pachner moves}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/573-unimodal-pachner-connectivity.md},
note = {Entry 573 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00913, "Unimodal connectivity by 2–3 and 3–2 Pachner moves", clopen.solutions.