ABSTRACT
Curves in space exhibit new types of global properties, in particular, knottedness and linking. Stokes’ Theorem presents a global property of curves in space in that it relates a quantity calculated along the curve with a quantity that depends on any surface that has that curve as a boundary. This chapter presents two main theorems: the Fary-Milnor Theorem, which shows how the property of knottedness imposes a condition on the total curvature of a curve, and Gauss’s formula for the linking number of two curves. The Fary-Milnor Theorem gives a necessary relationship between a knotted curve and the curvature of a space curve. Knottedness is a property that concerns how a simple closed curve “sits” in the ambient space. The notion of linking between two simple closed curves is a notion that considers how the curves are embedded in space in relation to each other.
