  We have developed a parallel algorithm that allows us to enumerate the number
of self-avoiding polygons on the square lattice to perimeter length 110. We
have also extended the series for the first 10 area-weighted moments and the
radius of gyration to 100. Analysis of the resulting series yields very
accurate estimates of the connective constant $\mu =2.63815853031(3)$ (biased)
and the critical exponent $\alpha = 0.5000001(2)$ (unbiased). In addition we
obtain very accurate estimates for the leading amplitudes confirming to a high
degree of accuracy various predictions for universal amplitude combinations.
