The Ideal Stratum, Ropelength Barriers, and Deformation Persistence of Knot and Link Types

We study path components of ropelength sublevel spaces for knot and link types. For an ordered oriented link type $\mathcal L$, let $Y_\Lambda(\mathcal L)$ be the moduli space of its $C^{1,1}$ representatives with standard thickness at least $1$ and total length at most $\Lambda$. The first nonempty level is the ropelength, and its minimizer locus is the ideal stratum $I(\mathcal L)$.
The first main result is a compactness theorem: if the graph of nonzero pairwise linking numbers of $\mathcal L$ is connected, then every $Y_\Lambda(\mathcal L)$ is compact in the quotient constant-speed $C^1$ topology. We combine this with a general minimax principle showing that distinct connected components of a compact minimizer locus are separated by a strictly positive energy barrier. Applied to the Gordian pair of Kusner and Kusner, this gives a rigorous nontrivial lower bound for the number of ideal deformation components: their two minimizing configurations lie in distinct ideal components and have strictly positive ropelength merge distance.
We also prove a general sink-to-birth principle for sublevel persistence. A non-global sink of an energy functional produces a path component born strictly above the global minimum. Bauermeister's construction of non-global ropelength sinks therefore implies that, in every two-component Gehring link-homotopy class, component persistence has a noninitial birth. Thus deformation persistence is not, in general, determined only by the ideal stratum and subsequent mergers.