cotton on vs dig

cotton on is not a noun while dig is a noun.

Word	Noun	Adjective	Verb	Adverb
cotton on	No	No	Yes	No
dig	Yes	No	Yes	No

As verbs, dig is a hypernym of cotton on; that is, dig is a word with a broader meaning than cotton on:

Other hypernyms of cotton on include apprehend, compass, comprehend, get the picture, grasp, grok, savvy.

cotton on (noun)	dig (noun)
	the act of digging
	the act of touching someone suddenly with your finger or elbow
	a small gouge (as in the cover of a book)
	an aggressive remark directed at a person like a missile and intended to have a telling effect
	the site of an archeological exploration

cotton on (verb)	dig (verb)
understand, usually after some initial difficulty	get the meaning of something
	turn up, loosen, or remove earth
	create by digging
	remove the inner part or the core of
	poke or thrust abruptly
	thrust down or into
	remove, harvest, or recover by digging
	work hard