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 $c(\tau)\le1$. 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 $c(\tau)\le1$. 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 ($r\ge4$), and each is determined up to $E(3)$ by its datum. No uniform quantitative form of this rigidity exists; but on each stratum $\Delta=\inf_s\sqrt{\tau^2+(\tau')^2}\ge\delta>0$ with uniform $C^5$ and curvature bounds the orbit distance obeys a log-Lipschitz bound, whose optimal constants diverge as $\delta\downarrow0$ 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.