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We survey and enlarge the known mappings of the 16-vertex model, withemphasis on mappings between the even and odd 8-vertex subcases of the generalmodel, also giving new mappings between these models, valid on finite toroidallattices. In particular, we find new mappings between the models by using theiralgebraic invariants with respect to the $SL(2)\times SL(2)$ symmetry of the16-vertex model; we also find a larger set of weak-graph transformations. Weshow many examples of models with negative weights which map to models withonly positive weights. Using the algebraic invariant relations of the even andodd 8-vertex models, we find the complete set of points in the complex fieldplane of the square lattice Ising model in a field which map to the even or odd8-vertex models; these points also correspond to the set of free-fermionicpoints of the model. We do not find any new integrable points, but we find anew mapping between the odd 8-vertex model and the square lattice Ising modelat magnetic field $H= i\pi/(2\beta)$, valid on finite toroidal lattices. Wealso show directly through various examples that mappings via algebraicinvariants do not fully exhaust the possible mappings a model may have withanother model. We construct a new solution to the odd 8-vertex free-fermionmodel which is valid on the finite lattice, since the previous known solutionresulted from a mapping valid only in the thermodynamic limit. Finally, wedetail for the first time the phase transitions of the column staggeredfree-fermion 8-vertex model, and show that it can be mapped to the bi-partitestaggered free-fermion model.

The 16-vertex model and its even and odd 8-vertex subcases on the square lattice