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Compressed Data Structures for Heegaard Splitting

Authors Henrique Ennes , Clément Maria

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LIPIcs.SoCG.2026.42.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Geometric topology
  • Theory of computation → Data structures design and analysis
Keywords
  • 3-manifold
  • Heegaard splitting
  • curves on surfaces
  • surface theory
  • data structure
  • computational topology

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Abstract

Heegaard splittings provide a natural representation of closed 3-manifolds by gluing handlebodies along a common surface. These splittings can be equivalently given by two finite sets of meridians lying on the surface, which define a Heegaard diagram. We present a data structure to effectively represent Heegaard diagrams as normal curves with respect to triangulations of a surface of complexity measured by the space required to express the normal coordinates' vectors in binary. This structure can be significantly more compressed than triangulations of 3-manifolds, giving exponential gains for some families. Even with this succinct definition of complexity, we establish polynomial-time algorithms for comparing and manipulating diagrams, performing stabilizations, detecting trivial stabilizations and reductions, and computing topological invariants of the underlying manifolds, such as their fundamental and homology groups. We also contrast early implementations of our techniques with standard software programs for 3-manifolds, achieving faster algorithms for the average cases and exponential gains in speed for some particular presentations of the inputs.

Cite As Get BibTex

Henrique Ennes and Clément Maria. Compressed Data Structures for Heegaard Splitting. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 42:1-42:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.42

Author Details

Henrique Ennes
  • Université Côte d'Azur, Nice, France
  • INRIA d'Université Côte d'Azur, Sophia Antipolis, France
Clément Maria
  • INRIA d'Université Côte d'Azur, Sophia Antipolis, France

Funding

This work has been partially supported by the ANR project ANR-20-CE48-0007 (AlgoKnot) and the project ANR-15-IDEX-0001 (UCA JEDI).
  • Ennes, Henrique: The PhD thesis, which this work is a part of, has been supported by the French government, through the France 2030 investment plan managed by the Agence Nationale de la Recherche, as part of the "UCA DS4H" project, reference ANR-17-EURE-0004.

Supplementary Materials

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