Papers, software, and other writing

Publications

Jack Brand, Benjamin A. Burton, Zsuzsanna Dancso, Alexander He, Adele Jackson, and Joan Licata. Arc diagrams on 3-manifold spines. Discrete & Computational Geometry, 2023.
DOI: 10.1007/s00454-023-00539-4
- Online access.
- Preprint (arXiv:2202.02007)
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)), Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2023: p. 21:1-21:16.
  DOI: 10.4230/LIPIcs.SoCG.2023.21
  UQ SMP Best Student Publication in Mathematics, 2023.
- Journal version:
  Preprint (arXiv:2303.06321).
- 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

Benjamin A. Burton, Thiago de Paiva, Alexander He and Connie On Yu Hui. Crushing surfaces of positive genus.
arXiv:2403.11523.
Alexander He, James Morgan and Em Thompson. An algorithm to construct one-vertex triangulations of Heegaard splittings.
arXiv:2312.17556 (v2).
Benjamin A. Burton and Alexander He. Connecting 3-manifold triangulations with monotonic sequences of elementary moves.
- Extended abstract:
  Presented at the Computational Geometry: Young Researchers Forum 2021 (CG:YRF 2021).
- Journal version:
  arXiv:2012.02398 (v2).

Regina: I have contributed some small routines, most notably:
1. 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.
2. Some new elementary moves for 4-manifold triangulations.
Counterexamples for triangulations:
Supporting code for the paper titled Finding large counterexamples by selectively exploring the Pachner graph. The main purpose is running a targeted search for counterexamples to a family of conjectures concerning 3-manifold triangulations.
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.
Knot decomposition (in development):
Implementation of an algorithm (designed in collaboration with Eric Sedgwick and Jonathan Spreer) for constructing "edge-ideal" triangulations of the prime summands of a knot.
Orbit-counting (in development):
Implementation of the orbit-counting algorithm introduced by Agol, Hass and Thurston.

Combinatorial transformations in 3-manifold topology (PhD thesis, undergoing examination)
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.
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.