Unsigned Frenet Data of Closed Space Curves: Exact Fibres, Generic Rigidity, and Conditional Stability

Abstract

A closed positively curved space curve is determined by its curvature and signed torsion up to an orientation-preserving rigid motion; that sign is the only place the ambient orientation enters. We ask what survives its loss, for closed embedded curves in ${\mathbb{R}}^3$ with $\kappa >0$ compared pointwise in a common arclength label. The answer is governed by the branch invariant $c(\tau )$, the number of components left by the infinite-order zero set of $\tau$: the smooth signed lifts of $|\tau |$ number exactly $2^{c(\tau )}$, and reduce to $\{\tau ,-\tau \}$ precisely when $c(\tau )\le 1$. Hence a given unsigned datum is carried by at most $2^{c(\tau )}$ classes modulo $SE(3)$, and by a single $E(3)$-orbit when $c(\tau )\le 1$. Both extremes occur: for arbitrary knot types $K_1,\dots ,K_m$ there is a datum with fibre exactly $2^m$ classes modulo $SE(3)$, realising all connected sums of the $K_i$ and their mirrors; under a chirality hypothesis these are $2^m$ knot types. Conversely, curves with only simple torsion zeros are open and dense, hence residual, among parametrised $C^r$ embeddings ($r\ge 4$), and each is determined up to $E(3)$ by its datum. No uniform quantitative form of this rigidity exists; but on each stratum $\Delta ={\inf }_s\sqrt{{\tau }^2+({\tau }^{\prime }{)}^2}\ge \delta >0$ with uniform $C^5$ and curvature bounds the orbit distance obeys a log-Lipschitz bound, whose optimal constants diverge as $\delta \downarrow 0$ on the strata containing a fixed exact ambiguous pair. The engine is a one-dimensional inverse estimate for the signed square root, logarithmically optimal at that level.

49 pages, 6 figures. v2 substantially revised: fibre counts separated, topological seed replaced by a formal local knot insertion lemma with proof, genericity completed, conditional stability for labelled orbit distance, near-collision sequences degenerate in Δ, constants explicit