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.
Crookedness
See also
Bridge IndexExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
CrookednessCite this as:
Weisstein, Eric W. "Crookedness." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Crookedness.html