 
Summary: RESEARCH BLOG 5/4/04
At the PNGS at University of Utah, Brian White talked about em
bedded curves with total curvature 4. By the F´ary Milnor theorem,
any such curve must be an unknot. In fact, there is a close relation
between the minimal curvature of a knot and its bridge number. The
bridge number is the minimal number of maxima over embeddings of
the knot in R3
. It's not hard to see that if the knot has bridge number
b, then the total curvature can be made arbitrarily close to 2b, since
one may connect 2b maxima and minima with curvature each by a
braid with strands with curvature arbitrarily close to 0. Conversely,
F´ary showed that the total curvature of a fixed curve in R3
is the av
erage over the total curvature over all projections of the curve to R2
(this is essentially Fubini's theorem). It's not hard to see that the total
curvature of a knot of bridge number b is at least 2b, since the projec
tion must have at least b maxima and b minima, and any arc between
a maximum and minimum must contribute at least to the curvature,
since it is infimized by the length of the image of the Gauss map to S1
.
