  Various interacting lattice path models of polymer collapse in two dimensions
demonstrate different critical behaviours. This difference has been without a
clear explanation. The collapse transition has been variously seen to be in the
Duplantier-Saleur $\theta$-point university class (specific heat cusp), the
interacting trail class (specific heat divergence) or even first-order. Here we
study via Monte Carlo simulation a generalisation of the Duplantier-Saleur
model on the honeycomb lattice and also a generalisation of the so-called
vertex-interacting self-avoiding walk model (configurations are actually
restricted trails known as grooves) on the triangular lattice. Crucially for
both models we have three and two body interactions explicitly and
differentially weighted. We show that both models have similar phase diagrams
when considered in these larger two-parameter spaces. They demonstrate regions
for which the collapse transition is first-order for high three body
interactions and regions where the collapse is in the Duplantier-Saleur
$\theta$-point university class. We conjecture a higher order multiple critical
point separating these two types of collapse.
