curl up vs curl

curl up is not a noun while curl is a noun.

Word	Noun	Adjective	Verb	Adverb
curl up	No	No	Yes	No
curl	Yes	No	Yes	No

As verbs, curl up and curl are synonyms defined as:

Other synonyms of curl up include draw in.

curl up (noun)	curl (noun)
	lock of hair in the shape of a spiral or curl
	a round shape formed by a series of concentric circles (as formed by leaves or flower petals)

curl up (verb)	curl (verb)
shape one's body into a curl	form a curl, curve, or kink
	play the Scottish game of curling
	twist or roll into coils or ringlets
	wind around something in coils or loops
	shape one's body into a curl