On Sparse Representations of 3-Manifolds
Krist\'of Husz\'ar, Cl\'ement Maria

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…
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Taxonomy
TopicsGeometric and Algebraic Topology · Topological and Geometric Data Analysis · Advanced Graph Theory Research
