Let $I=[0, l]$ be a closed interval, $k(s), \tau(s)$ smooth real valued functions on $I$ with $k$ strictly positive at all points, and $\mathbf{t}_{0}, \mathbf{n}_{0}, \mathbf{b}_{0}$ a positively oriented orthonormal triad of vectors in $\mathbf{R}^{3}$. An application of the fundamental theorem on the existence of solutions to ODEs implies that there exists a unique smooth family of triples of vectors $\mathbf{t}(s), \mathbf{n}(s), \mathbf{b}(s)$ for $s \in I$ satisfying the differential equations

$$\mathbf{t}^{\prime}=k \mathbf{n}, \quad \mathbf{n}^{\prime}=-k \mathbf{t}-\tau \mathbf{b}, \quad \mathbf{b}^{\prime}=\tau \mathbf{n}$$

with initial conditions $\mathbf{t}(0)=\mathbf{t}_{0}, \mathbf{n}(0)=\mathbf{n}_{0}$ and $\mathbf{b}(0)=\mathbf{b}_{0}$, and that $\{\mathbf{t}(s), \mathbf{n}(s), \mathbf{b}(s)\}$ forms a positively oriented orthonormal triad for all $s \in I$. Assuming this fact, consider $\alpha: I \rightarrow \mathbf{R}^{3}$ defined by $\alpha(s)=\int_{0}^{s} \mathbf{t}(t) d t$; show that $\alpha$ defines a smooth immersed curve parametrized by arc-length, which has curvature and torsion given by $k(s)$ and $\tau(s)$, and that $\alpha$ is uniquely determined by this property up to rigid motions of $\mathbf{R}^{3}$. Prove that $\alpha$ is a plane curve if and only if $\tau$ is identically zero.

If $a>0$, calculate the curvature and torsion of the smooth curve given by

$$\alpha(s)=(a \cos (s / c), a \sin (s / c), b s / c), \quad \text { where } c=\sqrt{a^{2}+b^{2}}$$

Suppose now that $\alpha:[0,2 \pi] \rightarrow \mathbf{R}^{3}$ is a smooth simple closed curve parametrized by arc-length with curvature everywhere positive. If both $k$ and $\tau$ are constant, show that $k=1$ and $\tau=0$. If $k$ is constant and $\tau$ is not identically zero, show that $k>1$. Explain what it means for $\alpha$ to be knotted; if $\alpha$ is knotted and $\tau$ is constant, show that $k(s)>2$ for some $s \in[0,2 \pi]$. [You may use standard results from the course if you state them precisely.]