Crookedness

Let a knot  be parameterized by a vector function  with , and let  be a fixed unit vector in . Count the number of local 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.

REFERENCES:

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

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, p. 115, 1976.