It is permitted to chew gum on Yom Kippur?
No, it is not permitted to chew gum on Yom Kippur.
Tasting and expelling food is not permitted on Yom Kippur (Orach Chaim 567), and chewing gum is similar, to a degree, to tasting and expelling. Yet, there is room to suggest that gum is different because it is never swallowed.
However, gum is in fact more stringent than tasting, because according to many authorities it is considered as a form of eating, and a person chewing gum has to recite a berachah before, so that it will be clearly prohibited to chew gum (see here).
It appears, however, that there shouldn’t be a prohibition against chewing already-chewed gum (meaning, gum that was chewed before Yom Kippur) — though I don’t know if anybody would want to do this on Yom Kippur (or any other time).