Compressed data structures for Heegaard splittings

2025-07-15

AI Summary

This paper addresses the problem of efficient representation and computation for Heegaard diagrams of 3-manifolds. Methodologically, it introduces a compact data structure based on normal curves in surface triangulations, employing binary normal coordinates to achieve exponential space compression while preserving polynomial-time complexity. The framework supports key operations—including diagram comparison via destabilization, stabilization detection, and rapid computation of topological invariants such as the fundamental group and first homology group (H_1). Its primary contribution is the first integration of normal curve theory with Heegaard splittings, yielding a unified framework that simultaneously ensures high compressibility and computational tractability. Experimental evaluation demonstrates that the method outperforms existing software on average, achieving exponential speedups on specific instances, thereby significantly enhancing both the efficiency and precision of topological analysis for 3-manifolds.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

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 in 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, given 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 first homology groups. We also contrast early implementations of our techniques with standard software programs for 3-manifolds, achieving better precision and faster algorithms for the average cases and exponential gains in speed for some particular presentations of the inputs.

Problem

• Efficiently represent Heegaard diagrams as compressed data structures
• Develop polynomial-time algorithms for comparing and manipulating diagrams
• Compute topological invariants of 3-manifolds with improved precision and speed

Innovation

• Compressed data structures for Heegaard diagrams
• Polynomial time algorithms for manifold operations
• Exponential speed gains in specific cases