  We determine lower bounds for the topological complexity of many planar
polygon spaces mod isometry. With very few exceptions, the upper and lower
bounds given by dimension and cohomology considerations differ by 1. This is
true for 130 of the 134 generic 7-gon spaces. Our results apply to spaces of
n-gons for all n, but primarily for those whose genetic codes, in the sense of
Hausmann and Rodriguez, are moderately small.
