Is evolute unique for a space curve? The book by T.J. Willmore states that for a space curve there are infinitely many involutes. But it emphasizes again and again that for any of the infinitely many involutes the given curve is the evolute. Even the internet says that evolute is unique. I'm confused if evolute is unique or not for a space curve?

There are, in fact, always infinitely many _involutes_ of a given space curve.

However, only for a plane curve can you say that the evolute is unique. All the helices on a cylinder of fixed radius have the same involute, so they are all evolutes of the same curve.