  Recently, the method of one-dimensional maps was introduced as a means of
generating exceptional discretisations of the $\phi^4$-theories, i.e., discrete
$\phi^4$-models which support kinks centred at a continuous range of positions
relative to the lattice. In this paper, we employ this method to obtain
exceptional discretisations of the sine-Gordon equation (i.e. exceptional
Frenkel-Kontorova chains). We also use one-dimensional maps to construct a
discrete sine-Gordon equation supporting kinks moving with arbitrary velocities
without emitting radiation.
