Crookedness

Let a Knot $K$ be parameterized by a Vector Function ${\bf v}(t)$ with $t\in \Bbb{S}^1$, and let ${\bf w}$ be a fixed Unit Vector in $\Bbb{R}^3$. Count the number of Relative Minima of the projection function ${\bf w}\cdot{\bf v}(t)$. Then the Minimum such number over all directions ${\bf w}$ and all $K$ of the given type is called the crookedness $\mu(K)$. Milnor (1950) showed that $2\pi\mu(K)$ is the Infimum of the total curvature of $K$. For any Tame Knot $K$ in $\Bbb{R}^3$, $\mu(K)=b(K)$ where $b(K)$ is the Bridge Index.

See also 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.