  In the usual statistical model of a dense polymer (a single space-filling
loop on a lattice) in two dimensions the loop does not cross itself. We modify
this by including intersections in which {\em three} lines can cross at the
same point, with some statistical weight w per crossing. We show that our model
describes a line of critical theories with continuously-varying exponents
depending on w, described by a conformally-invariant non-linear sigma model
with varying coupling constant g_\sigma^2 >0. For the boundary critical
behavior, or the model defined in a strip, we propose an exact formula for the
\ell-leg exponents, h_\ell=g_\sigma^2 \ell(\ell-2)/8, which is shown
numerically to hold very well.
