Originally Posted by David Lyga
A razor will most likely not work on glass. The carbide cutter is harder than the glass, a razor is not. What you want is not a cut in the glass, but a scratch. After scratching a line along a steel ruler or other a straight-edge, I place a dowel (and you would need a thin one) directly under the scratch, a piece of paper towel over the glass and gently press down on each side of the dowel. When I'm lucky, the glass parts at the line of the scratch, when I'm not, I have to clean up the little uneven bits with a tile nibbler or glass pliers (used by stained-glass workers). A carborundum sharpening stone works as well.
Many commonly available glass cutters have a sort of bulb at the end of the handle. I watched a master stained-glass worker cut intricate designs by using that steel ball to tap on the underside of the glass along the scratch just before he broke the piece free. He said that the vibrations from the tapping actually cracked the glass along the line, making even curved cuts come out clean. I have also used the bottle-cutting trick of heating the glass gently with a torch or a flame and then rubbing an ice cube along the line. It's over-kill for what you want to do, but it works for me in difficult situations.