Subdivision of Bézier curves for ambient isotopy in molecular modeling

Knot Morphing Algorithm for Quantum `Fragile Topology'

“…Moreover, after sufficiently many iterations, the Bézier curves are ambient isotopic to the initial control polygon [12,14,15]. It has been shown [12] that iterations 0 through 3 produce unknotted Bézier curves, while the 4th iteration results in the Bézier curve being 4 1 , as shown in Figure 1(b).…”

“…These perturbations range over all vectors in R 3 , while the methods presented here restrict new control points to a given polygon. Equivalence of a Bézier curve to a given piecewise linear knot has been established under a convergent sequence [15], with sufficient [14] and necessary conditions [12] for some knots, by contributions of two of the present authors.…”