  We study two simple modifications of self-avoiding polygons. Osculating
polygons are a super-set in which we allow the perimeter of the polygon to
touch at a vertex. Neighbour-avoiding polygons are only allowed to have nearest
neighbour vertices provided these are joined by the associated edge and thus
form a sub-set of self-avoiding polygons. We use the finite lattice method to
count the number of osculating polygons and neighbour-avoiding polygons on the
square lattice. We also calculate their radius of gyration and the first
area-weighted moment. Analysis of the series confirms exact predictions for the
critical exponents and the universality of various amplitude combinations. For
both cases we have found exact solutions for the number of convex and
almost-convex polygons.
