Computing braids from approximate data

TL;DR

This paper develops methods for computing braids from approximate path data, addressing numerical instability in exact algorithms by formalizing an input model based on separation predicates, applicable to polynomial root tracking.

Contribution

It introduces a formal input model for approximate data using separation predicates, enabling stable braid computation from uncertain path descriptions.

Findings

Formalized an input model for approximate path data.

Connected certified polynomial root tracking to braid computation.

Addressed numerical instability in braid algorithms.

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.