Compressed data structures for Heegaard splittings

TL;DR

This paper introduces a compact data structure for representing Heegaard diagrams of 3-manifolds, enabling efficient algorithms for topological computations and outperforming existing software in speed and size.

Contribution

The paper presents a novel compressed data structure for Heegaard diagrams that allows polynomial-time algorithms for key topological operations, with exponential size reductions in certain cases.

Findings

Significantly more compact representation than traditional triangulations.

Polynomial-time algorithms for diagram comparison and manipulation.

Faster average-case performance and exponential speedups in specific scenarios.

Abstract

Heegaard splittings provide a natural representation of closed 3-manifolds by gluing two 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, where the complexity is measured by the space required to express the normal coordinates' vectors in binary. This structure can be significantly more compact than triangulations of 3-manifolds, yielding exponential gains for certain 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.