Topological complexity (within 1) of the space of isometry classes of   planar n-gons for sufficiently large n
  Hausmann and Rodriguez classified spaces of isometry classes of planar n-gons according to their genetic code, which is a collection of sets (called genes) containing n. Omitting the n yields what we call gees. We prove that, for a set of gees with largest gee of size k>0, the topological complexity (TC) of the associated space of n-gons is either 2n-5 or 2n-6 if n>2k+2. We present evidence that suggests that it is very rare that the TC is not equal to 2n-5 or 2n-6. 
Donald M. Davis
math.AT math.CO