On Sparse Representations of 3-Manifolds

TL;DR

This paper explores algorithms for transforming 3-manifold triangulations to control sparsity parameters like treewidth and edge valence, enabling efficient computation of quantum invariants and simplified Heegaard diagrams.

Contribution

It introduces algorithms that preserve or control sparsity measures during triangulation transformations, facilitating efficient topological computations.

Findings

Treewidth is preserved during triangulation to Heegaard diagram conversion.

A quasi-linear-time algorithm reduces maximum edge valence to nine.

Combined algorithms produce simplified Heegaard diagrams with limited intersections.

Abstract

3-manifolds are commonly represented as triangulations, consisting of abstract tetrahedra whose triangular faces are identified in pairs. The combinatorial sparsity of a triangulation, as measured by the treewidth of its dual graph, plays a fundamental role in the design of parameterized algorithms. In this work, we investigate algorithmic procedures that transform or modify a given triangulation while controlling specific sparsity parameters. First, we revisit a standard, linear-time algorithm that converts a given triangulation into a Heegaard diagram of the underlying 3-manifold, showing that the construction preserves treewidth. We apply this construction to exhibit a fixed-parameter tractable framework for computing Kuperberg's quantum invariants of 3-manifolds. Second, we present a quasi-linear-time algorithm that retriangulates a given triangulation into one with maximum edge valence of at most nine, while only moderately increasing the treewidth of the dual graph. Combining these two algorithms yields a quasi-linear-time algorithm that produces, from a given triangulation, a Heegaard diagram in which every attaching curve intersects at most nine others.