  We note that two formulations of dual gonihedric Ising models in 3d, one
based on using Wegner's general framework for duality to construct a dual
Hamiltonian for codimension one surfaces, the other on constructing a dual
Hamiltonian for two-dimensional surfaces, are related by a variant of the
standard decoration/iteration transformation.
  The dual Hamiltonian for two-dimensional surfaces contains a mixture of link
and vertex spins and as a consequence possesses a gauge invariance which is
inherited by the codimension one surface Hamiltonian. This gauge invariance
ensures the latter is equivalent to a third formulation, an anisotropic
Ashkin-Teller model. We describe the equivalences in detail and discuss some
Monte-Carlo simulations which support these observations.
