Benjamin A. Burton, Thiago de Paiva, Alexander He and Connie On Yu Hui. Crushing surfaces of positive genus. Algebraic & Geometric Topology, 2025.
DOI: 10.2140/agt.2025.25.4949
Alexander He, James Morgan and Em K. Thompson. An algorithm to construct one-vertex triangulations of Heegaard splittings. Journal of Computational Geometry, 2025.
DOI: 10.20382/jocg.v16i1a18
Alexander He, Eric Sedgwick and Jonathan Spreer. A practical algorithm for knot factorisation.
Conference version:
In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025).
DOI: 10.4230/LIPIcs.SoCG.2025.55
Full version:
Preprint (arxiv:2504.03942v1).
Benjamin A. Burton and Alexander He. Connecting 3-manifold triangulations with unimodal sequences of elementary moves. Discrete & Computational Geometry, 2025.
DOI: 10.1007/s00454-025-00735-4
Extended abstract:
Presented at the Computational Geometry: Young Researchers Forum 2021 (CG:YRF 2021).
Jack Brand, Benjamin A. Burton, Zsuzsanna Dancso, Alexander He, Adele Jackson, and Joan Licata. Arc diagrams on 3-manifold spines. Discrete & Computational Geometry, 2024.
DOI: 10.1007/s00454-023-00539-4
Benjamin A. Burton and Alexander He. Finding large counterexamples by selectively exploring the Pachner graph.
Conference version:
In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023).
DOI: 10.4230/LIPIcs.SoCG.2023.21
UQ SMP Best Student Publication in Mathematics, 2023.
Full version:
Preprint (arXiv:2303.06321v2).
Extended abstract:
In the Oberwolfach Report for Oberwolfach Workshop 2303 – Low-dimensional topology.
DOI: 10.14760/OWR-2023-3
Benjamin A. Burton and Alexander He. On the hardness of finding normal surfaces. Journal of Applied and Computational Topology, 2021.
DOI: 10.1007/s41468-021-00076-0
Alexander He, Eric Sedgwick, and Jonathan Spreer. A practical algorithm for recognising Seifert fibred spaces with nonempty boundary.
In preparation.
Natalie DiMichele, Michael Forbes, Alexander He, and Benjamin A. Burton. Certifiable Optimisation: A Framework with Applications to Normal Surface Detection.
Submitted.
Triangulations in Geometry and Topology (Dagstuhl Seminar 24072)
DOI: 10.4230/DagRep.14.2.120
By Maike Buchin, Jean Cardinal, Arnaud de Mesmay, Jonathan Spreer, Alex He and all authors of the abstracts in this report
Regina: I have contributed some new algorithms and constructions, most notably:
Handlebody recognition for 3-manifold triangulations, which is described in detail as an auxiliary algorithm in the paper titled Finding large counterexamples by selectively exploring the Pachner graph.
Some new elementary moves for 4-manifold triangulations.
Algorithms using edge-ideal triangulations:
Implementation of algorithms developed in joint work with Eric Sedgwick and Jonathan Spreer. See the papers titled A practical algorithm for knot factorisation and A practical algorithm for recognising Seifert fibred spaces with nonempty boundary (working title, in preparation).
A version of the code which only implements the knot factorisation algorithm has been archived at the following Software Heritage permalink:
https://archive.softwareheritage.org/swh:1:dir:54d0e58d0977b955e152398a1797d63938787670
Counterexamples for triangulations:
Supporting code for the paper titled Finding large counterexamples by selectively exploring the Pachner graph, which is joint work with Ben Burton. The main purpose is running a targeted search for counterexamples to a family of conjectures concerning 3-manifold triangulations.
Software Heritage permalink: https://archive.softwareheritage.org/swh:1:dir:5aca986985119792687eb6fab54857c986e93e32
Constructing one-vertex triangulations from Heegaard diagrams:
Implementation of the main algorithm from the paper titled An algorithm to construct one-vertex triangulations of Heegaard splittings, which is joint work with James Morgan and Em Thompson.
Software Heritage permalink: https://archive.softwareheritage.org/swh:1:dir:4365ee594746fdd427bb1ce580bd9e2e7f051d11
Orbit-counting (in development):
Implementation of the extended orbit-counting algorithm introduced by Agol, Hass and Thurston.
Combinatorial transformations in 3-manifold topology (PhD thesis)
DOI: 10.14264/3df9845
For the most part, this PhD thesis compiles results from various papers. Some of the exposition has been updated, and in some places the thesis contains more proof details than what appears in the corresponding paper.
I am currently aware of a handful of errors in this thesis. These are either typos, or minor mathematical omissions that do not affect the correctness of the main results.
Computational complexity of problems in normal surface theory (Honours thesis)
The main results from this Honours thesis appear in the paper titled On the hardness of finding normal surfaces, so you should probably look at that paper instead. The other material in this thesis consists mostly of: (1) standard background material (for which there exist many far better sources), and (2) proof ideas that failed.
If you are still determined to look at this thesis, be warned that some of the typesetting is terrible. I am also aware of at least one typo; at some point I might come back and add a list of errors.