Questions about torsion of a curve in $\mathbb{R}^3$ and analogues of torsion in higher dimensions Suppose we have a curve $\alpha(s) : I \to \mathbb{R}^3$ parametrized by arc-length that has nowhere-vanishing second derivative, so that we are able to define the torsion $\tau(s)$ for every $s \in I$. It is clear to me that one can deform a curve in a way that can change the curvature by large amounts while keeping $\tau = 0$ everywhere. I think that it is possible to deform a curve in a way that we can vary torsion while keeping curvature fixed. I was wondering if anybody could give me a nice example of this.
My second question involves extending the idea of torsion. As far as I understand, torsion measures the planarity of a curve. For a curve that is embedded in $\mathbb{R}^n$ is there a quantity indicating how far the curve is from being embedded in $\mathbb{R}^{n-1}$, and is this even useful?
 A: 
I think that it is possible to deform a curve in a way that we can vary torsion while keeping curvature fixed. I was wondering if anybody could give me a nice example of this.

Arguably the simplest class of examples are those for which the curvature $\kappa$ and torsion $\tau$ are constants for each curve in the family. Any such curve is a helix (including the degenerate case of zero torsion, which gives a circle) and can be parameterized by
$$\alpha(t) := (r \cos t, r \sin t, bt)$$
for some parameters $r$ (the radius of the unique cylinder containing the image of the helix) and $b$ (a quantity which controls the component velocity in the direction of the axis of that cylinder). (Computing gives that $||\alpha'(t)||^2 = \sqrt{r^2 + b^2}$, and in particular $\alpha$ is a constant-speed parameterization, so it's no trouble to write down an arc length parameterization.) Now, direct computation gives that the curvature $\kappa$ and torsion $\tau$ of $\alpha$ are
$$\kappa(t) = \frac{r}{r^2 + b^2} \qquad \text{and} \qquad \tau(t) = \frac{b}{r^2 + b^2}.$$
So, to produce a family of helices $\color{#bf0000}{\alpha_m(t)}$ with constant prescribed curvature $\kappa$ and varying torsion $\tau$, we need only pick (nonconstant) functions $r(m), b(m)$ that satisfy $$\kappa = \frac{r(m)}{r(m)^2 + b(m)^2}$$ for any prescribed constant $\kappa > 0$. Rearranging shows that this equation defines a circle
$$\left(r - \frac{1}{2 \kappa}\right)^2 + b^2 = \left(\frac{1}{2 \kappa} \right)^2$$
in $rb$-space, and we can produce explicit functions $r(m), b(m)$ by parameterizing this circle (or more precisely, this circle less the point $(r, b) = (0, 0)$). The usual rational parameterization of the unit circle, for example, leads to the solution
\begin{align}
r(m) := \frac{1}{(1 + m^2) \kappa} \\
b(m) := \frac{m}{(1 + m^2) \kappa}
\end{align}
and hence to the parameterized family of helices defined by
$$\color{#bf0000}{\alpha_m(t) = \left(\frac{\cos t}{(1 + m^2) \kappa}, \frac{\sin t}{(1 + m^2) \kappa}, \frac{m t}{(1 + m^2) \kappa}\right)}.$$ Substituting this parameterization in the above formulas for $\kappa$ and $\tau$ reveals that
$$m = \frac{\tau}{\kappa};$$
in particular, when $\kappa = 1$ the parameter $m$ is nothing more than the torsion $\tau$ itself.
This animation shows how $\color{#bf0000}{\alpha_m}$ (with $\kappa = 1$) varies with prescribed torsion $\tau$.

This was generated by the following Maple code:
with(plots):
stau := [cos(t) / (tau^2 + 1), sin(t) / (tau^2 + 1), tau * t / (1 + tau^2)];
opts := color=black, numpoints = 400:
animate(spacecurve, [stau, t = -64..64, opts], tau = -4..4, view = [-2..2, -2..2, -8..8], scaling = constrained, frames = 192, axes = none);

One can generalize this example wildly, by the way, as given any functions $\kappa(s)$, $\tau(s)$ (say, with $\kappa > 0$) parameterized by arc length there is a curve with curvature $\kappa(s)$ and torsion $\tau(s)$, and this curve is unique up to Euclidean motions, though actually solving for such a curve requires integrating a differential equation.


For a curve that is embedded in $\Bbb R^n$ is there a quantity indicating how far the curve is from being embedded in $\Bbb R^n$, and is this even useful?

Yes, there are higher-order analogues of torsion for Euclidean spaces $\Bbb R^n$, $n > 3$. Recall that in in $\Bbb R^3$ (1) one can always choose (at least for curves with nonvanishing curvature) a unique adapted orthonormal frame $({\bf T}, {\bf N}, {\bf B})$ along a given smooth curve, and (2) derivatives of  the curvature and torsion satisfy
$$\begin{pmatrix} {\bf T} \\ {\bf N} \\ {\bf B} \end{pmatrix}' = \begin{pmatrix} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0\end{pmatrix}\begin{pmatrix} {\bf T} \\ {\bf N} \\ {\bf B} \end{pmatrix}.$$ (Here, $'$ denotes differentiation w.r.t. an arc length parameter.) Similarly, in $\Bbb R^4$, for sufficiently generic curves one can choose a unique adapted orthonormal frame $({\bf T}, {\bf N}, {\bf B}, \color{#0000ff}{{\bf U}})$ (here $\color{#0000ff}{{\bf U}}$ is sometimes called, predictably, the trinormal), and such a curve has three curvature quantities, $\kappa, \tau, \color{#00bf00}{\upsilon}$, and these satisfy
$$\begin{pmatrix} {\bf T} \\ {\bf N} \\ {\bf B} \\ \color{#0000ff}{{\bf U}} \end{pmatrix}' = \begin{pmatrix} 0 & \kappa & 0 & 0\\ -\kappa & 0 & \tau & 0 \\ 0 & -\tau & 0 & \color{#00bf00}{\upsilon} \\ 0 & 0 & -\color{#00bf00}{\upsilon} & 0\end{pmatrix}\begin{pmatrix} {\bf T} \\ {\bf N} \\ {\bf B} \\ \color{#0000ff}{{\bf U}} \end{pmatrix}.$$ As you guessed, $\color{#00bf00}{\upsilon}$ measures the failure of the curve to be contained in the hyperplane $\langle {\bf T}, {\bf N}, {\bf B} \rangle$ to the appropriate order. (Preserving the analogy with the $3$-dimensional case, the triple $(\kappa, \tau, \color{#00bf00}{\upsilon})$ is a complete set of invariants for a generic curve in $\Bbb R^4$, in that their values generically determine a curve uniquely up to Euclidean motions.) The analogous statements for higher dimensions are the generalizations from the $\Bbb R^3$ and $\Bbb R^4$ cases that you'd guess; in particular generic curves in $\Bbb R^n$ have, and are generically determined by, $n - 1$ curvature functions. (Note that this general pattern captures the $2$-dimensional case, too, in which there is only a single invariant, just the usual (signed) curvature.)
