The Ideal Stratum and Deformation Persistence of Knot Types

We introduce a deformation-persistence framework for knot types based on normalized spaces of representatives. For a knot type K and a parameter Lambda >= 0, let Y_Lambda(K) be the space of representatives with thickness at least 1 and length at most Lambda, modulo reparametrization and orientation-preserving Euclidean isometries. Admissible deformations are paths staying inside some Y_Lambda(K), and their path components define admissible components. The first nonempty level occurs at the ropelength Rop(K), and the corresponding minimizer locus is called the ideal stratum. We define ideal admissible components, their number, and ideal merge scales, which record when distinct ideal components become admissibly connected as the length bound is relaxed. These merge scales induce a finite-valued ultrapseudometric on the set of ideal components. We then construct the associated pure merge Vietoris--Rips filtration, a simplicial encoding of the zero-dimensional merge persistence. Finally, we discuss finite-dimensional polygonal approximations as a computational model for constrained knot-shape spaces.