The Ideal Stratum and Deformation Persistence of Knot Types

TL;DR

This paper introduces a new deformation-based framework for analyzing knot types using ropelength and persistence concepts, revealing the structure of minimal ropelength representatives.

Contribution

It defines the ideal stratum and a ropelength ultrapseudometric, linking minimal ropelength shapes to a persistence theory and providing new invariants for knot classification.

Findings

The first birth level of the admissible-component persistence equals the ropelength.

The ropelength ultrapseudometric satisfies the strong triangle inequality.

The pure merge Vietoris--Rips filtration encodes the merge data without higher homological content.

Abstract

We study a knot type through the ropelength-filtered spaces of its thick representatives. For a knot type $K$ and a scale parameter $\Lambda>0$, let $Y_\Lambda(K)=\mathcal{R}_{1,\Lambda}(K)$ be the space of representatives of $K$ with thickness at least $1$ and length at most $\Lambda$, modulo reparametrization and orientation-preserving Euclidean isometries. The basic equivalence relation is defined by admissible deformations: two representatives are equivalent at scale $\Lambda$ if they can be joined through representatives that remain in $Y_\Lambda(K)$. The resulting admissible components form a one-parameter persistence object as $\Lambda$ 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 $\Pi^{\mathrm{ad}}_{\mathrm{ideal}}(K)$, the ideal component number $\nu_{\mathrm{ideal}}(K)$, and pairwise ideal merge scales. For the fixed knot type $K$, the central invariant introduced here is the ropelength ultrapseudometric $d_{\mathrm{merge}}(C,D)=\mu_{\mathrm{ideal}}(C,D)-\operatorname{Rop}(K)$ defined for $C,D\in\Pi^{\mathrm{ad}}_{\mathrm{ideal}}(K)$. 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.