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In the usual statistical model of a dense polymer (a single space-fillingloop on a lattice) in two dimensions the loop does not cross itself. We modifythis by including intersections in which {\em three} lines can cross at thesame point, with some statistical weight w per crossing. We show that our modeldescribes a line of critical theories with continuously-varying exponentsdepending on w, described by a conformally-invariant non-linear sigma modelwith varying coupling constant g_\sigma^2 >0. For the boundary criticalbehavior, 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 shownnumerically to hold very well.

Universality classes of dense polymers and conformal sigma models