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"My only concern is that it wouldn't work, which I see as a problem."
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Hot-wire foam cutters work by running electric current through a nichrome (or sometimes stainless steel) wire to heat it
up. Then foam is pressed against the wire, and the wire melts the foam, forming a cut. In the process, the wire gets
cooled by heat being transferred to the foam. This limits the
cutting speed. If you turn up the current, the sections of
wire not in contact with the foam get even hotter, and waste more power.
Another point is that nichrome has a positive temperature coefficient of resistivity: its resistivity is higher at higher
temperatures. Therefore, the section of wire cooled by cutting the foam has a lower resistivity than the section not
cutting the foam. At first glance, it would seem like this would help, by allowing more current and thus dissipating more
power in that section, but it's actually a hindrance, because the current through the whole length of wire is necessarily
the same. Therefore, the hotter sections, with higher resistance, actually dissipate more power, and therefore get
hotter for a given current.
This idea seeks to counteract that. Simply take a wire made of a material with a negative temperature coefficient
(greater in magnitude than that of nichrome), and wrap that in nichrome. Then draw it to the desired size. This similar
to making Wollaston wire.
By putting the nichrome on the outside, its convenient property of forming a passivation layer when exposed to oxygen
at high temperature is retained.
Now, when a section of wire gets cooled, the internal material gets more resistive, forcing more of the current through
the outer nichrome layer. When a section of wire gets hotter, the internal material gets less resistive, allowing more
current to avoid going through the nichrome.
The result is a heating wire that is locally thermostatic at all points along its length. It should maintain a roughly
constant temperature along its whole length for any given current.
Balancing the resistivities, temperature coefficients, and cross-sectional areas of the two materials is left as an exercise
for the reader.
Other potential applications are heated clothing, and home heating hairs [link].
54/366 [shortly after April 2017]
Potential application: home heating hairs
My big warm hairy house
Mentioned in idea body [notexactly, Nov 12 2018]
 Nichrome data
[notexactly, Nov 13 2018]
 Table of resistivities and temperature coefficients
[notexactly, Nov 13 2018]
||This makes sense, but (a) how significant is the positive
coefficient effect for nichrome and (b) are there suitable
materials with a negative coefficient?
||a) According to the data that Wikipedia has (which comes
from what looks from the name in the citation to be a
wire manufacturer's website), nichrome increases in
resistance 3.3% from 20 to 315 °C.  Wikipedia also says
its temperature coefficient is 0.0004 /K. 
||b) According to , amorphous carbon, germanium, or
silicon might work. Maybe also the materials used in NTC
thermistors (though I don't know what those are made
||I thought of another possible application: Wire-loop
desoldering of SMD ICs, for which I cannot find a video,
but which is normally done with hot air and an unheated
wire. You'd have to make sure the voltage gradient along
the wire is low enough to be safe for the chip and/or the
rest of the circuit (depending on which you're trying to
save), though. Maybe it would work for hot-bar soldering
too, but that seems to work fine with current technology,
because the bar is thick enough to distribute heat evenly.
||Going with the Wollaston wire idea, wouldn't it make more sense to focus on thermal conductivity?
||You need a way of redistributing the heat rapidly from the hotter to the cooler areas. You need a core with relatively high, constant electrical resistance, but the lowest possible thermal resistance.
||Those two physical characteristics may not be easy to reconcile, though.
||Graphene has very high thermal conductivity in the plane of
its sheets, but very low perpendicular to the plane. I expect
carbon nanotubes, being graphene rolled into tubes, would
have very high thermal conductivity along their axes, but
low radially. Either of those could probably be doped to
increase electrical resistance.
||Isn't heating chaos? Should not technology be trying to find the perfect electromagnetic, kinetic pattern to part the sea of foam and not have any macroscopic heat at all?
||I like macroscopic heat. It keeps me alive.
||^ Taking that way of thinking the project shed is going to be pretty toasty.