We form negative declarative clauses with not after be (she is not talking), after modal verbs (they must not go) and after auxiliary verbs do and have (we did not like it; they have not eaten). Cambridge Dictionary
And I found the following sentence in Oxford Dictionary:
perhaps not surprisingly, he was cautious about committing himself.
But perhaps is not a helping verb. So, Can we use "not" without helping verb?
The Negation Rule: In English, in order to claim that something is not true, you form a negative sentence by adding the word not after the first auxiliary verb in the positive sentence. If there is no auxiliary verb in the positive sentence, as in the Present Simple and Past Simple tenses, then you add one (in both these cases, the auxiliary verb do). White Smoke
We make negatives by putting not after the first part of the verb. British Council
Another example of using not without an auxiliary verb:
I can think of a hundred reasons not to come.
is the above negative clauses? if yes, What is the rule to use not without auxiliary verb? if no, what is the part of speech of not in the above sentence?