Can one ritually slaughter with a wire? Why or why not? The Talmud says that any implement may be used, so according to that a wire would be fine. Are there any technical problems that would invalidate a wire?
No, a wire would not be a good shechita knife, because when it's cutting, it goes entirely under the neck, which is a problem of חלדה, "tunneling." חלדה is one of the five main halachos of shechita. See שמלה חדשה 24:9 et seqq for more info.
To borrow an illustration of חלדה from a previous answer of mine, this is not חלדה, but if the circled area goes under the skin, that is חלדה. There is no way to cut with a wire without the wire being buried under the skin of the neck.
(click for larger image; image not to scale)
Additionally, like 6591 points out, דרסה will be an issue here; I don't see a way to use a wire to cut without using excessive downward force.
Drisa is the problem that comes to mind.
A sharp object that cuts by slicing when moved across the animals neck is allowed.
Drisa is cutting through pressure applied in a downward force. Using a cheese cutter or the like would render the animal a niveila. Unfit to eat and Tamei.
see https://en.wikipedia.org/wiki/Shechita under The five forbidden techniques when using a halaf (sakin) to slaughter animals
the only problem that i see that is hard to avoid is (from above)
חלדה Haladah (digging or burying) - The knife must be drawn over the throat so that it is visible while shechita is being performed. It must not be stabbed into the neck or buried by fur, hide, or feathers in the case of a bird. Haladah occurs if the shochet either accidentally cuts into the animal's throat so deeply that the entire width of the knife disappears in the wound, uses a knife that is too short so that the end disappears in the wound, or if a foreign object falls over the knife so the shochet loses sight of the incision.
and you will need to hold the string "knife" with both hands since the use of something like the frame of the "Fret Saw" might also be considered Haladah since the string will not be seen when looking at if from the side of the frame
PS also there might be a problem of it not being considered a knife as simla chadosho 8.4 were it can be understood that a knife as short as the width of a pigeons neck is not considered a knife and even if you do the cutting motion it is considered that you are making a hole