This talk is about smooth closed curves in two and three dimensions.
Our main goal will be to investigate "total curvature" of such loops. For signed curvature in $\mathbb R^2\!,$ this yields the "rotation number" of our loop, about which we will mention some nifty results. Taking absolute curvature, we get a quantity that is floppier, but has interesting lower bounds.
Here, we take an interlude to discuss a length formula from geometric measure theory. This will give us a glimpse into the behavior of short curves on spheres and spaghetti thrown against a wall.
With these tools, we show that any loop in $\mathbb R^3$ has total curvature $\geq 2\pi$, with equality if and only if the curve is planar and convex (Fenchel). Finally, we show that any knotted loop in $\mathbb R^3$ has total curvature $\geq 4\pi$ (Fáry-Milnor).