{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/an-algorithm-to-construct-one-vertex","title":"An algorithm to construct one-vertex triangulations of Heegaard splittings","arxiv_id":"2312.17556","date":"2023-12-29","proceeding":null,"authors":["Alexander He","James Morgan","Em K. Thompson"],"abstract":"Following work of Jaco and Rubinstein (2006), which (non-constructively) proved that any 3-manifold admits a one-vertex layered triangulation, we present an algorithm, with implementation using Regina, that uses a combinatorial presentation of a Heegaard diagram to construct a generalised notion of a layered triangulation. We show that work of Husz\\'ar and Spreer (2019) extends to our construction: given a genus-$g$ Heegaard splitting, our algorithm generates a triangulation with cutwidth bounded above by $4g-2$. Beyond Heegaard splittings, our construction actually extends to a natural generalisation of Dehn fillings: given a one-vertex triangulation with a genus-$g$ boundary component $B$, we can construct a one-vertex triangulation of any 3-manifold obtained by filling $B$ with a handlebody. To demonstrate the usefulness of our algorithm, we present findings from preliminary computer searches using this algorithm.","url_abs":"https://arxiv.org/abs/2312.17556v2","url_pdf":"https://arxiv.org/pdf/2312.17556v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"an-algorithm-to-construct-one-vertex","repo_url":"https://github.com/alexhe98/heegaardbuilder","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}