As far as I know, only two things are capable of cutting diamonds. The first is the only material strong enough: another diamond. The second is high energy in a precision beam: a laser.
Aside from that, I dont know with any certainty. I wouldnt doubt it if chemistry could be used to dissolve away at the carbon that comprises a diamond, but I dont think it would be very precise or pretty, and I am no chemist to begin with.
