Computing braids from approximate data

TL;DR

The paper tackles the problem of deriving combinatorial braids from approximate root trajectories by replacing fragile lexicographic-ordered representations with a separation-predicate–based input model. It introduces arrangements as open, convex cells in the complex configuration space and uses permutation points to extract braids from sequences of arrangements, enabling robust braid computation from approximate data produced by certified path tracking. The authors present a complete implementation framework featuring a pointed-arrangement data structure, dynamic topological sorts for updates, and an on-the-fly braid accumulation algorithm, with termination and correctness guarantees under bounded numerical uncertainty. This approach bridges geometric braid data with computational braid invariants, supporting applications in braid monodromy and the topology of plane curves, and is accompanied by a practical Rust implementation that relies on interval arithmetic for sep queries.

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.

Figures (9)

• Figure 1: Path in $C_n$ inducing the combinatorial braid $\sigma_i$.
• Figure 2: Illustration of Algorithm \ref{['algo:cover']} for four points moving along a circle, with position known only approximately (as represented by the blue discs). The bars represent the lifetime of an edge in the graph representing the current arrangement. The diagrams in the last row represent the resulting cover. The arrangement cell marked with $*$ is not representable as a configuration of boxes, it is defined by $\left\{ \max(\operatorname{Re}(z_2), \operatorname{Re}(z_3)) < \min(\operatorname{Re}(z_1), \operatorname{Re}(z_4)), \operatorname{Im}(z_3) < \operatorname{Im}(z_2) \text{ and } \operatorname{Im}(z_4) < \operatorname{Im}(z_1) \right\}.$
• Figure 3: All four combinatorial arrangements of two axis-aligned disjoint rectangles in the plane. Two configurations are equivalent if, for every pair of rectangles, the same relative position holds in both configurations (namely, whether one lies strictly to the left of the other or strictly above the other).
• Figure 4: All 40 combinatorial arrangements of three axis-aligned disjoint rectangles in the plane.
• Figure 5: All 772 combinatorial configurations of disjoint 4 rectangles in the plane.

Theorems & Definitions (21)

• Definition 1: Path approximation
• Definition 2: Path approximation, computational variant
• Definition 3: Arrangement, Arrangement cell
• Lemma 4
• proof
• Definition 5: Cover
• Lemma 6
• proof
• Proposition 7
• proof