(I present four theories, in rough reverse order of when I thought of them. Theory 0 is my current best guess and may actually be right; I think it fits all the information OP has given us in the riddle and in subsequent hints. But it's a little bit artificial somehow. The other theories handle the riddle itself reasonably well, in some cases a bit better than theory 0, but fall down on the latest hint.)
Theory 0
Our object is
1. Only one thigh: yes, there's just a linear sequence of arrows. 2. Removing neck or thigh still leaves the $A\rightarrow B$ arrow there and tells us something about it. 3. Yes, this is algebra. 4. If we don't do any slicing we have an exact sequence $0\rightarrow A\rightarrow B\rightarrow 0$ which implies that the $A\rightarrow B$ arrow is an isomorphism ("invertible").
Theory 1
Our object is
one function/morphism/map/arrow in an exact sequence. This time the head end is the end where the arrowheads are.
Slice my neck, that's quite epic.
Cut off the sequence at the "head" end and put a 0 in place instead, so we have $\cdots\rightarrow A\rightarrow B\rightarrow 0$. This means that the $A\rightarrow B$ morphism is epic.
Slice my thigh, that's quite demonic.
Cut off the sequence at the "foot" end and put a 0 in place instead, so we have $0\rightarrow A\rightarrow B\rightarrow\cdots$. This means that the $A\rightarrow B$ morphism is monic.
Slice my waist, you shall have a kill.
If we cut between A and B, then the morphism we're looking at just isn't there at all. It's been killed.
Slice all three, that's an overkill.
If you do all three slices, then you not only destroy the morphism we're looking at, you also cut off some other things that you didn't need to. Overkill, indeed.
Theory 2
Alternatively, maybe our object is
a short exact sequence; that is, some mathematical objects and morphisms $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ with the property that the image of each is the kernel of the next. We again adopt the convention that the "head" end is the right, the direction in which the arrow-heads point. So A, B, C are the thigh, the waist, and the neck.
Slice my neck, that's quite epic.
This means: replace C (the neck) with 0. We then have $0\rightarrow A\rightarrow B\rightarrow 0$ and what this tells us is that $A\rightarrow B$ is epic.
Slice my thigh, that's quite demonic.
This means: replace A (the thigh) with 0. We then have $0\rightarrow B\rightarrow C\rightarrow 0$ and what this tells us is that $B\rightarrow C$ is monic.
Slice my waist, you shall have a kill.
Now we replace B with 0 instead, getting $0\rightarrow A\rightarrow 0\rightarrow C\rightarrow 0$, which forces both A and C to be 0. We've killed them.
Slice all three, that's an overkill.
... And now we have $0\rightarrow 0\rightarrow 0\rightarrow 0\rightarrow 0$, which in one sense means we've killed everything but in another tells us nothing at all. "Overkill" seems fair.
Theory 3
Changing focus a little, maybe our object is
not a mathematical object but a bit of related mathematical _notation; specifically, an arrow $\rightarrow$
and the slicing is
done by adding extra arrowheads or, in one case, slashes.
Slice my neck, that's quite epic.
Once again, the head end is at the right next to the arrow-head. So we get $\twoheadrightarrow$ which is used in mathematics to denote a function that is epic.
Slice my thigh, that's quite demonic.
Now we get $\rightarrowtail$ which is used in mathematics to denote a function that is monic.
Slice my waist, you shall have a kill.
$\not\rightarrow$ which isn't used much but clearly means that there isn't a map/function/arrow of some sort, which one could reasonably call "a kill". I don't recall ever actually seeing this used, but TeX and its mathematical fonts have specific support for writing it (compare the start of this paragraph with e.g. what you get by applying \not to other varieties of arrow: $\not\twoheadrightarrow$), which suggests that it is a thing people sometimes want.
Slice all three, that's an overkill.
In the sense that no one would in fact ever write $\rlap{\not\rightarrowtail}\twoheadrightarrow$, which is altogether too much of a good thing.
Other notes
In category theory (the branch of mathematics where this sort of diagram most natively dwells) there are a number of things whose names could suggest killing: ends, limits, terminal objects. None of them seems to fit this riddle, though. There's also a thing called a "Killing vector field" (named after someone called Killing) but again nothing about it other than the name seems to fit.