Let a knot K be parameterized by a vector function v(t) with t in S^1, and let w be a fixed unit vector in R^3. Count the number of local minima of the projection function w·v(t). Then the minimum such number over all directions w and all K of the given type is called the crookedness mu(K). Milnor (1950) showed that 2pimu(K) is the infimum of the total curvature of K. For any tame knot K in R^3, mu(K)=b(K) where b(K) is the bridge index.

See also

Bridge Index

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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.

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Cite this as:

Weisstein, Eric W. "Crookedness." From MathWorld--A Wolfram Web Resource.

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