The Ideal Stratum and Deformation Persistence of Knot Types

Abstract

We study a knot type through the ropelength-filtered spaces of its thick representatives. For a knot type $K$ and a scale parameter $Λ>0$, let $Y_Λ(K)=\mathcal{R}_{1,Λ}(K)$ be the space of representatives of $K$ with thickness at least $1$ and length at most $Λ$, modulo reparametrization and orientation-preserving Euclidean isometries. The basic equivalence relation is defined by admissible deformations: two representatives are equivalent at scale $Λ$ if they can be joined through representatives that remain in $Y_Λ(K)$. The resulting admissible components form a one-parameter persistence object as $Λ$ increases. We prove that the first birth level of this admissible-component persistence is exactly the ropelength $\operatorname{Rop}(K)$. The initial layer $I(K)=Y_{\operatorname{Rop}(K)}(K)$ is the ideal stratum of $K$. Thus the ropelength-minimizing locus is not treated merely as a set of ideal shapes, but as the birth stratum of a constrained deformation theory. We define the ideal admissible-component set $Π^{\mathrm{ad}}_{\mathrm{ideal}}(K)$, the ideal component number $ν_{\mathrm{ideal}}(K)$, and pairwise ideal merge scales. For the fixed knot type $K$, the central invariant introduced here is the ropelength ultrapseudometric $d_{\mathrm{merge}}(C,D)=μ_{\mathrm{ideal}}(C,D)-\operatorname{Rop}(K)$ defined for $C,D\inΠ^{\mathrm{ad}}_{\mathrm{ideal}}(K)$. We prove that this function is finite-valued and satisfies the strong triangle inequality. The pure merge Vietoris--Rips filtration is a secondary simplicial encoding of this ultrapseudometric structure: it records the same zero-dimensional merge data and has no higher-dimensional homological content beyond the merge partition itself. We also compute the basic case of the unknot and indicate finite polygonal and diagrammatic approximations as further directions.