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I am going to try a mix of two of the methods. If I use a glass cutter and some support in order to score a horizontal line around the cutting line, but THEN tie the string with oil around the bottle right on top of the scored line, the fire should do its job, but the break should be smoother, along the scored line.
To smooth glass, use sandpaper. Glass is really soft and wears away quickly when rubbed with sand paper. The thing is, if it has sharp edges, (which it does, that's why you are sanding it!) it can cut through the sandpaper and cut your hand. I would recommend gluing the sandpaper to a little block of wood. Work slowly, and protect your hands and eyes.