  By performing Monte Carlo sampling of $N$-steps self-avoiding polygons
embedded on different Bravais lattices we explore the robustness of
universality in the entropic, metric and geometrical properties of knotted
polymer rings. In particular, by simulating polygons with $N$ up to $10^5$ we
furnish a sharp estimate of the asymptotic values of the knot probability
ratios and show their independence on the lattice type. This universal feature
was previously suggested although with different estimates of the asymptotic
values. In addition we show that the scaling behavior of the mean squared
radius of gyration of polygons depends on their knot type only through its
correction to scaling. Finally, as a measure of the geometrical
self-entanglement of the SAPs we consider the standard deviation of the writhe
distribution and estimate its power-law behavior in the large $N$ limit. The
estimates of the power exponent do depend neither on the lattice nor on the
knot type, strongly supporting an extension of the universality property to
some features of the geometrical entanglement.
