Almost every mention of bottle cutting includes a reference to using a hot wire to crack the glass, usually with very few details. Usually mentioned are a low voltage power supply and nichrome wire, sometimes with a reference to foam cutters. The simple facts are that the wire has got to be used like the string (above) to stress the glass and either run the crack along the score or crack the glass locally without a score.
The wire must wrap the bottle to stress it all at once because glass cracks are notorious for wandering off to the side, ruining the even edge. This is quite unlike a foam cutter which uses a wire under tension to make progressive cuts like a hack, band, or coping saw blade.
The low voltage power supply is possible, a fairly heavy battery like a car battery probably being required, as a couple of flashlight batteries, as used in miniature scenery cutters, will not last long with a longer wire.
The low voltage supply ideally should be adjustable but it does not have to be DC so a household doorbell transformer, if of enough wattage would work. An ideal tool is a Variac, a variable transformer with a dial. Many variable voltage power supplies sold for electronics work may not have enough amperage.
The problem as I see it is that the wire must wrap around the bottle, bottles come in different sizes so the wrap must also, nichrome wire becomes stiff and brittle when heated, and if an even break is to occur, the wrap must pass close to itself, possibly creating a short circuit across a fairly high amperage supply..