  We extend the tangle model, originally developed by Ernst and Sumners, to
include composite knots. We show that, for any prime tangle, there are no
rational tangle attachments of distance greater than one that first yield a
4-plat and then a connected sum of 4-plats. This is done by building on results
on exceptional Dehn fillings at maximal distance. We then apply our results to
the action of the Hin recombinase on mutated sites. In particular, after
solving the tangle equations for processive recombination, we use our work to
give a complete set of solutions to the tangle equations modelling distributive
recombination.
