One of the main invariants associated with the study of smooth curves is the total curvature. It is defined by the line integral of the curvature function on the curve itself.

We understand that a closed curve in the space is unknotted if it can be deformated to a circle. More precisely, if there exists a homeomorphism from the disk into the space, R^3, such that the trace´s curve is the image of the boundary of the disk. 

An interesting fact about knots is the Fary-Milnor theorem. It gives us a limit on the total curvature of an knot, namely, if a curve 𝞪 is a knot then its total curvature is greater than 4𝝅.

In the next animation we present some well-known knots for which we have computed their total curvature using Maple.