Well, auxiliary verbs can take "not". Main verbs cannot. The thing is, "have" can work as a main verb or an auxiliary verb.
When you see "have not/haven't", it's usually an auxiliary verb, and another verb follows it to show the main action (see Maddie's examples). You see this in present perfect grammar.
When you see "do not have/don't have", you understand "have" as the main action.