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The relevance of crossings at the collapse point is not clear. While the cubic perturbation is still believed to be relevant at the tri-critical $O(n)$ point \cite{Nienhuis_2008}, a recent numerical study \cite{Bedini_2013} seems to indicate that the Duplantier and Saleur universality class is stable in the presence of crossings, at least with respect to the cross-over and length-scale exponents.  Very recently Nahum {\it et al.\ }\cite{Nahum_2013} published a study of loop models with crossings. Their analysis is based on a replica limit of the $\sigma$ model on real projective space $\mathbb{RP}^{n-1}$. They give a field theoretic description of the ISAT which explains the phase diagram found numerically in \cite{Foster_2009} and suggests that the ISAT collapse transition is an infinite-order multi-critical point.\subsection{A model with competing interactions}  In this paper we consider a polymer model of self-avoiding trails with both multiply-visited site and nearest-neighbour interactions. This model generalises ISAT and INNSAT. It also contains in a limiting case the ISAW model, when the Boltzmann weight associated with multiply visited sites is sent to zero.  We study the model via computer simulations using the flatPERM algorithm, and so extend the study of INNSAT in \cite{Bedini_2013}. We point out that this model has been studied some time ago by Wu and Bradley \cite{Wu_1990} via real-space renormalisation, which predicted a tetra-critical point   separating the ISAT and ISAW collapse points. In contrast, we find that there is likely to be the ISAT collapsed point itself, that separates a line of first-order transitions from ISAW-like weaker $\theta$-point type transitions.  In \ref{sec:isat-nn} we define the model introduced by Wu and Bradley. In Section 3, we present the results of  our simulational studies and deduce a conjectured phase diagram. We end by summarising our conclusions in Section 5.