Yesterday, I was thinking about creating a function that distinguishes between different topological knots (embeddings of $S^1$ into $\mathbb R^3$) and I came up with the following function $f$. If $K$ is some knot embedded in $\mathbb R^3$, I defined $f(K)$ as
$$f(K)=\min_{K^*\cong K} \int_{K^*} \kappa \space d\vec{l}$$
where $K^*$ is some $C^2$ curve in $\mathbb R^3$ that can be continuously deformed to $K$, and $\kappa$ is the curvature of $K^*$ at a point (sorry, I realize all of the k’s can get confusing). Basically, all this is doing is integrating over the curvature of the knot at each point. I like this integral because dilating $K^*$ shouldn’t actually affect the integral’s value, because stretching by a factor of $\alpha$ increases the length of the knot by a factor of $\alpha$ but decreases the curvature at each point by a factor of $\alpha$, which cancels out.
Let $U$ be the unknot and $T$ be the trefoil knot. My intuition tells me that
$$f(U)=2\pi$$
and
$$f(T)=4\pi$$
because the unknot can be represented as a plain circle, and the integral in question will always equal $2\pi$ when integrating over a circle (since the radius of curvature is constant). For the trefoil knot, it doesn’t quite look like a circle, but we can deform it so it looks just like a circle that wraps around twice:
But the tricky part is making this rigorous and proving that this arrangement actually does minimize the curvature. My intuition tells me that it does by “eliminating unnecessary curves,” but I’m not sure how to make a rigorous proof out of this. Am I correct, and if so, how can I rigorously prove these special values of $f(K)$?
More generally, is there a straightforward method to calculate $f(K)$ for more complicated knots?
Milnor calculates that this is $2\pi$ times an integer invariant he calls the crookedness of a knot. The crookedness of a non-trivial knot is greater than 1, which proves that $f(T) = 4\pi$, to use your notation, or more generally that for any non-trivial knot $K$ we have $f(K) \geq 4\pi$. This is called the Fary-Milnor theorem.
See his paper "On the total curvature of knots"