  An exact calculation of the phase diagram for a loop gas model on the
brickwork lattice is presented. The model includes a bending energy. In the
dense limit, where all the lattice sites are occupied, a phase transition
occuring at an asymmetric Lifshitz tricritical point is observed as the
temperature associated with the bending energy is varied. Various critical
exponents are calculated. At lower densities, two lines of transitions (in the
Ising universality class) are observed, terminated by a tricritical point,
where there is a change in the modulation of the correlation function. To each
tricritical point an associated disorder line is found.
