How about a plasma cutter? I know a plasma cutter will cut through normal silica. (I cut through some steel plate laying right on top of large gravel.) Silicon dioxides melting point is 1700 C, and aluminum oxide is 2100 C, a little higher.
I doubt this material is impervious to shear force, or being lowered in temperature until it's brittle and cracked. But those aren't methods of cutting.
What I was really getting at though is that if you look at the definition of "cut" you'll see in most (all?) sources it defines it has being achieved through a sharp object or tool. I understand that's not often how it's used colloquially though.