What is the grammaticality of "not" in "I hope not" and "I know not"?

This question stemmed from the lyrics of the song Vague Utopia by Tia Gostelow. One of the lines from the chorus is as following:

I just want you to know,
And i know not what the younger dreams,
Some vague utopia, utopia

The structure I know not what the… sounded a bit weird to me. After doing some research, the closest similar grammatically valid structures(from oxford) that I found are:

1. Treating the symptoms and not the cause. (negative of the cause).
2. How was it? Not so bad. (Not very close to the actual lyrics).

I also think the lyrics could be using archaic English, when I know not, I run not etc were permissible in English. Please explain if the lyrics is grammatically correct, and if yes, what rule is it using.

Grazie