Recently, the method of one-dimensional maps was introduced as a means of generating exceptional discretisations of the $��^4$-theories, i.e., discrete $��^4$-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.

20 pages