Cotton to is a older expression that I think is most often used in the south ( of the US). It means to find acceptable. " we don't cotton to your kind around here" meaning "people like you are not wanted here". I'm not familiar with cotton with or cotton up.
Wow. Thank you guys! I will simply remove this expression from my note and forget about it.
It went like this. I came across the expression "to cotton on to A" in a book written by a Scottish historian. The part was about American Civil War and words like "cotton industry" and "cotton bond" were everywhere in the page, but this one was obviously used as a verb. As always when I learn a new expression like this, I was mildly amused and perplexed at the same time. My dictionary had multiple entries under the word cotton, like "cotton to", "cotton up" etc. So, I just wanted to know if this usage was common or not. The sentence was like this: "When the French cottoned on to the insincerity of official German pledges to fulfill their reparations commitments, they drew the conclusion that reparations would have to be collected by force and invaded the industrial Ruhr region."