cut down vs knock off

cut down and knock off both are verbs.

Word	Noun	Adjective	Verb	Adverb
cut down	No	No	Yes	No
knock off	No	No	Yes	No

As verbs, knock off is a hyponym of cut down; that is, knock off is a word with a more specific, narrower meaning than cut down:

cut down (verb)	knock off (verb)
cut down on; make a reduction in	stop pursuing or acting
intercept (a player)	get rid of (someone who may be a threat) by killing
cause to come or go down	write quickly
cut with a blade or mower	take by theft
cause to fall by or as if by delivering a blow	cut the price of
cut with sweeping strokes; as with an ax or machete