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Practical Software for Triangulating and Simplifying 4-Manifolds

Author Rhuaidi Antonio Burke

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Author Details

Rhuaidi Antonio Burke
  • The University of Queensland, Brisbane, Australia

Funding

The author was supported by an Australian Government Research Training Program Scholarship.

Acknowledgements

We thank the referees for their helpful comments.

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Rhuaidi Antonio Burke. Practical Software for Triangulating and Simplifying 4-Manifolds. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 29:1-29:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.29

Abstract

Dimension 4 is the first dimension in which exotic smooth manifold pairs appear - manifolds which are topologically the same but for which there is no smooth deformation of one into the other. Whilst smooth and triangulated 4-manifolds do coincide, comparatively little work has been done towards gaining an understanding of smooth 4-manifolds from the discrete and algorithmic perspective. In this paper we introduce new software tools to make this possible, including a software implementation of an algorithm which enables us to build triangulations of 4-manifolds from Kirby diagrams, as well as a new heuristic for simplifying 4-manifold triangulations. Using these tools, we present new triangulations of several bounded exotic pairs, corks and plugs (objects responsible for "exoticity"), as well as the smallest known triangulation of the fundamental K3 surface. The small size of these triangulations benefit us by revealing fine structural features in 4-manifold triangulations.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Mathematics of computing → Mathematical software
Keywords
  • computational low-dimensional topology
  • triangulations
  • 4-manifolds
  • exotic 4-manifolds
  • mathematical software
  • experiments in low-dimensional topology

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