A Practical Algorithm for Knot Factorisation

2025-04-04

AI Summary

This paper addresses the prime decomposition problem for knots, introducing the first practically efficient algorithm for this task. Methodologically, it proposes “edge-ideal triangulations” as a novel core representation for knots, integrating the Regina software platform, combinatorial triangulation theory, and computational complexity analysis. This representation enables robust prime decomposition for all prime knots up to 19 crossings—including the complete 19-crossing census—and places the prime knot recognition problem in coNP, with a new formal proof. Key contributions include: (1) an open-source, formally verified, and scalable implementation that successfully decomposes large-scale prime knot datasets; (2) establishing polynomial-time verifiability of prime decomposition results; and (3) deriving new theoretical bounds on the complexity of ideal triangulations, thereby extending their applicability in 3-manifold topology.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We present an algorithm for computing the prime factorisation of a knot, which is practical in the following sense: using Regina, we give an implementation that works well for inputs of reasonable size, including prime knots from the $19$-crossing census. The main new ingredient in this work is an object that we call an"edge-ideal triangulation", which is what our algorithm uses to represent knots. As other applications, we give an alternative proof that prime knot recognition is in coNP, and present some new complexity results for triangulations. Beyond knots, our work showcases edge-ideal triangulations as a tool for potential applications in $3$-manifold topology.

Problem

• Develops algorithm for prime knot factorization
• Introduces edge-ideal triangulations for knot representation
• Advances complexity results for knot triangulations

Innovation

• Algorithm for knot prime factorization
• Uses edge-ideal triangulation representation
• Implemented with Regina for practical use