  We study a quantum chain of spin-1/2 particles whose world lines form a loop
gas. In addition to the usual interaction that projects a pair of neighboring
spins onto quantum-group spin 0, we introduce a three-site interaction that
projects onto spin 3/2. Changing the relative strengths of these interactions
leads to a rich phase diagram containing gapped and critical phases, and
exactly solvable points. For loop weight $n=2 \cos [\pi/(p+1)]$, the model
corresponds to N=p-3 species of interacting anyons, of which only the Fibonacci
case N=1 was studied previously. Our study applies to real n, and provides
further exact results which were unnoticed even for the special case N=1.
