You can build a hot wire cutter very easily. All you need is any old adapter.. a 12-19v laptop adapter +2A works great. Even a 9V battery will work for the hand tool (although not for the wide wire below, you need more V and A to smoothly cut that)
Basically get a stick of wood and then make a Y on with the springiness of a bow. Then get some nichrome wire (you can buy it at most hardware stores) and then basically clip some croc clips to the bars of the Y and off yu go.
This works great for me:
I can also use those clips to power other "toolings" like this board with adjustable heights by Screwing those wing nuts. This allows me to cut a surface to an exact thickness or even an incline section by setting the wingnuts to different heights.
You should also have some sort of spring to buffer the tension so you don't snap the wire. Here I am using a bit of old bike tyre, which works great. Some heavy duty elastic or bungie would be great too.
On a final note.. of course "soldering iron rules" apply, so advise getting one of those rotary timer switches to be safe, and like with my design there is a very bright LED that lets me know it is on.
PS this can also cut pretty much any thermoplastic, lego, foam etc.. some slower than others but you can adjust the heat easily by sliding a clip along the wire. For the hand tool as the wire is short i have a basic rheostat (not shown as it is on the other side), but is basically a coil of wire i can move the clip along if i need it cooler, so it doesnt smoke. Basically if it is smoking then it is too hot.. not unsafe, other than some nasty fumes, but you get a nasty "plastic hair" situation and molten plastic which can ruin a smooth cut).
