Computing braids from approximate data

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.

💡 Research Summary

The paper addresses the problem of converting a geometric braid—defined by n points moving continuously in the complex plane without collisions—into a combinatorial braid element of Artin’s braid group, when the input data are only available approximately. Classical exact algorithms rely on the lexicographic ordering of point coordinates to detect when two strands exchange positions. This ordering is highly unstable under numerical errors, making it unsuitable for data produced by certified homotopy continuation or other numerical path‑tracking methods that provide only approximate tubular neighborhoods or Taylor‑model representations of the root paths.

To overcome this limitation the authors introduce an abstract “separation predicate” sep(i, j, t) that, given a pair of strands i and j and a time t, returns a future time t′ ≥ t + ε and an axis q∈{Re, Im} together with possibly swapped indices (i′, j′). The guarantee is that for all s∈