  The double copy relates quantities in gauge, gravity and related theories. A
well-known procedure for relating exact classical solutions is the Weyl double
copy in four spacetime dimensions, and a three-dimensional analogue of this --
the Cotton double copy -- has recently been found for topologically massive
gauge theory and gravity. In this paper, we use twistor methods to provide a
derivation of the position-space Cotton double copy, where this is seen to
arise from combining appropriate data in so-called minitwistor space. Our
methods rely on a massive generalisation of the Penrose transform linking
spacetime fields with cohomology classes in minitwistor space. We identify the
relevant transform from the twistor literature, but also show that it naturally
arises from considering scattering amplitudes in momentum space. We show that
the Cotton double copy in position space is only valid for type N solutions,
but that a simple twistor space double copy is possible for non-type N
solutions, where we use anyons to illustrate our arguments.
