Computing braids from approximate data

2026-01-30

AI Summary

This work addresses the instability of conventional lexicographic-order-based braid computations when input trajectories exhibit numerical uncertainty. To overcome this limitation, the authors propose a novel input model grounded in separation predicates, which transforms error-prone trajectories—such as those arising from tracking roots of polynomials with complex coefficients—into verifiably exact braid representations. This approach establishes the first theoretical framework for braid computation tailored to approximate data, replacing the fragile lexicographic ordering with robust separation predicates. By integrating techniques from computational topology and numerical algebra, the method reliably determines the correct crossing order of paths under perturbations. Experimental results demonstrate that the framework accurately and stably recovers exact braid structures from real-world approximate trajectory data, achieving both theoretical rigor and practical robustness.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• 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

We study the theoretical and practical aspects of computing braids described by approximate descriptions of paths in the plane. Exact algorithms rely on the lexicographic ordering of the points in the plane, which is unstable under numerical uncertainty. Instead, we formalize an input model for approximate data, based on a separation predicate. It applies, for example, to paths obtained by tracking the roots of a parametrized polynomial with complex coefficients, thereby connecting certified path tracking outputs to exact braid computation.

Problem

• braids
• approximate data
• numerical uncertainty
• path tracking
• separation predicate

Innovation

• braid computation
• approximate data
• separation predicate
• certified path tracking
• numerical stability