On Sparse Representations of 3-Manifolds

2025-12-05

AI Summary

Algorithmic efficiency bottlenecks in triangulating 3-manifolds hinder practical computation of quantum invariants. Method: We introduce a parameterized optimization framework leveraging combinatorial sparsity—specifically, the treewidth of the dual graph. Our approach includes (i) the first linear-time algorithm converting a triangulation to a Heegaard diagram while preserving treewidth; (ii) a sparsification-based retriangulation strategy guaranteeing maximum edge degree ≤ 9; and (iii) joint optimization integrating treewidth analysis, Heegaard diagram construction, and edge-degree control. Contribution/Results: We achieve a quasilinear-time transformation from arbitrary triangulations to low-crossing Heegaard diagrams, substantially improving fixed-parameter tractability and computational efficiency of Kuperberg’s quantum invariant. This work establishes the first treewidth-driven, structurally informed algorithmic foundation for efficient quantum computation in 3-dimensional topology.

Technology Category

• Search and Optimization: Heuristic Search
• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• 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

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 describe a 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.

Problem

• Converting triangulations to Heegaard diagrams preserving treewidth
• Retriangulating to limit edge valence while controlling treewidth increase
• Enabling fixed-parameter tractable computation of quantum invariants for 3-manifolds

Innovation

• Converts triangulations to Heegaard diagrams preserving treewidth
• Retriangulates to limit edge valence while controlling treewidth growth
• Enables fixed-parameter tractable computation of quantum invariants