Self-avoiding polygons on the square lattice

We have developed an improved algorithm that allows us to enumerate the num ber of self-avoiding polygons on the square lattice to perimeter length 90. Analysis of the resulting series yields very accurate estimates of the con nective constant mu = 2.638 158 529 27(1) (biased) and the critical exponen t alpha = 0.500 0005(10) (unbiased). The critical point is indistinguishabl e from a root of the polynomial 581x(4) + 7x(2) - 13 = 0. An asymptotic exp ansion for the coefficients is given for all n. There is strong evidence fo r the absence of any non-analytic correction-to-scaling exponent.