Hi, from what I remember chewing gum was unfortunently a no no on this diet. I do not think it will knock you out of ketosis if it is sugar free, but it will make you hungry again, and you will be like starting all over again, so I did not risk it.
I am sure that one of the experts will be along soon, who could answer you properly, or let you know if I am wrong, but resist it for now until you can find out for sure.
Gray