  We present analyses of substantially extended series for both interacting
self-avoiding walks (ISAW) and polygons (ISAP) on the square lattice. We argue
that these provide good evidence that the free energies of both linear and ring
polymers are equal above the theta-temperature, extending the application of a
theorem of Tesi et. al. to two dimensions. Below the $\theta$-temperature the
conditions of this theorem break down, in contradistinction to three
dimensions, but an analysis of the ratio of the partition functions for ISAP
and ISAW indicates that the free energies are in fact equal at all temperatures
to at least within 1%. Any perceived difference can be interpreted as the
difference in the size of corrections-to-scaling in both problems. This may be
used to explain the vastly different values of the cross-over exponent
previously estimated for ISAP to that predicted theoretically, and numerically
confirmed, for ISAW. An analysis of newly extended neighbour-avoiding SAW
series is also given.
