Crookedness

Let a Knot be parameterized by a Vector Function with , and let be a fixed Unit Vector in . Count the number of Relative Minima of the projection function . Then the Minimum such number over all directions and all of the given type is called the crookedness . Milnor (1950) showed that is the Infimum of the total curvature of . For any Tame Knot in , where is the Bridge Index.

Milnor, J. W. On the Total Curvature of Knots.'' Ann. Math. 52, 248-257, 1950.