  We consider the expected value for the total curvature of a random closed
polygon. Numerical experiments have suggested that as the number of edges
becomes large, the difference between the expected total curvature of a random
closed polygon and a random open polygon with the same number of turning angles
approaches a positive constant. We show that this is true for a natural class
of probability measures on polygons, and give a formula for the constant in
terms of the moments of the edgelength distribution.
  We then consider the symmetric measure on closed polygons of fixed total
length constructed by Cantarella, Deguchi, and Shonkwiler. For this measure, we
are able to prove that the expected value of total curvature for a closed n-gon
is exactly \pi/2 n + (\pi/4) 2n/(2n-3). As a consequence, we show that at least
1/3 of fixed-length hexagons and 1/11 of fixed-length heptagons in 3-space are
unknotted.
