Generators, relations and symmetries in pairs of 3×3 unimodular matrices

Denote the free group on two letters by and the -representation variety of by . There is a -action on the coordinate ring of , and the geometric points of the subring of invariants is an affine variety . We determine explicit minimal generators and defining relations for the subring of invariants and show is a degree 6 hyper-surface in mapping onto . Our choice of generators exhibit symmetries which allow for a succinct expression of the defining relations.