poleaxe vs cut down

poleaxe is a noun but cut down is not a noun.

Word	Noun	Adjective	Verb	Adverb
poleaxe	Yes	No	Yes	No
cut down	No	No	Yes	No

As verbs, cut down is a hypernym of poleaxe; that is, cut down is a word with a broader meaning than poleaxe:

Other hypernyms of poleaxe include drop, fell, strike down.

poleaxe (noun)	cut down (noun)
a battle ax used in the Middle Ages; a long handled ax and a pick
an ax used to slaughter cattle; has a hammer opposite the blade

poleaxe (verb)	cut down (verb)
fell with or as if with a poleax	cut down on; make a reduction in
	intercept (a player)
	cause to come or go down
	cut with a blade or mower
	cause to fall by or as if by delivering a blow
	cut with sweeping strokes; as with an ax or machete