AI 中文总结
本文研究无符号Frenet数据对应的曲线纤维性质，构造出达到紧界的无符号数据，证明挠率仅具简单零点的曲线由无符号数据唯一确定，揭示了一般刚性的定量二分法及相关阻碍。
AI 中文摘要
具有正曲率的闭合空间曲线，由其曲率与带符号挠率唯一确定，仅差一个刚体运动。挠率的符号是环境定向进入Frenet数据的唯一位置，也是在无定向读取数据时最先丢失的信息。本文研究该信息丢失后保留的内容：对于ℝ³中曲率κ>0的闭合嵌入曲线，所有性质由一个整数c(τ)决定，即挠率τ在无穷阶零点集合的补集的连通分支数。|τ|的光滑带符号提升恰好有2^{c(τ)}个，当且仅当c(τ)≤1时，这些提升会简化为{τ,-τ}；因此，沿公共弧长标签携带(κ,|τ|)的曲线纤维，最多有2^{c(τ)}个全等类，当c(τ)≤1时为单个镜像对，两种极端情况均存在。对于任意m≥1及任意纽结类型K₁,…,Kₘ，本文构造了一个无符号数据，其纤维恰好由2ᵐ个全等类组成，实现了所有K_i^{±1}的连通和，说明该界是紧的。反之，挠率仅具有简单零点的曲线在参数化Cʳ嵌入空间（r≥4）中是开且稠密的，每个此类曲线由其无符号数据唯一确定，仅差O(3)变换。这种一般刚性不存在统一的定量形式，其阻碍定位于Δ=infₛ√(τ²+(τ')²)处。在每个满足Δ≥δ>0的层上，在一致C⁵及曲率界条件下，模E(3)的轨道距离最多为Cρ(‖(κ_α,τ_α²)-(κ_β,τ_β²)‖_{C⁰})，其中ρ(t)=t(1+log₊1/t)，而当δ→0时最优常数发散。核心机制是一维的：带符号函数可从其平方中以统一的对数-Lipschitz模从C⁰空间恢复到L¹空间，且无法去掉对数项。
英文摘要
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 $κ>0$ compared pointwise in a common arclength label. The answer is governed by the branch invariant $c(τ)$, the number of components left by the infinite-order zero set of $τ$: the smooth signed lifts of $|τ|$ number exactly $2^{c(τ)}$, and reduce to $\{τ,-τ\}$ precisely when $c(τ)\le1$. Hence a given unsigned datum is carried by at most $2^{c(τ)}$ classes modulo $SE(3)$, and by a single $E(3)$-orbit when $c(τ)\le1$. 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 $Δ=\inf_s\sqrt{τ^2+(τ')^2}\geδ>0$ with uniform $C^5$ and curvature bounds the orbit distance obeys a log-Lipschitz bound, whose optimal constants diverge as $δ\downarrow0$ 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.
Comments49 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
