When you want to convey the idea of purposely avoiding eye contact or not paying attention to someone or something, there are several synonyms you can use for the phrase "not look at." For example, you could say "avert one's gaze," "ignore," "turn a blind eye," "avoid eye contact," "not acknowledge," "disregard," "overlook," "shun," "refuse to see," "look away," "brush off," or "give the cold shoulder." Each of these phrases has a slightly different connotation, so it's worth considering which one best suits the context in which you want to use it.