Topological complexity of some planar polygon spaces

Let M ¯ n , r denote the space of isometry classes of n -gons in the plane with one side of length r and all others of length 1, and assume that 1 ≤ r < n - 3 and n - r is not an odd integer. Using known results about the mod-2 cohomology ring, we prove that its topological complexity satisfies TC ( M ¯ n , r ) ≥ 2 n - 6 . Since M ¯ n , r is an ( n - 3 ) -manifold, TC ( M ¯ n , r ) ≤ 2 n - 5 . So our result is within 1 of being optimal.