I slice mine all the time underwater. Done it about 20 time so far. I use a razor bade and make it qwik.
I used to think you were supposed to slice a branch off, but now I simply do it like a hair cut, leaving probably less than a qtr inch on the rock. It comes back very well.
Hopefully your fish didn't try to eat some. I believe Garf recommends a turkey baster at the ready to grab it when cut so it or pieces dont' get away from you for fish food (?) or getting stuck behind a rock where you don't want new polyps to grow.