Well, we know don't is the same as do not, right?
Therefore, can I say "Do not you know?", instead of "Don't you know?"?
Well, I know that chances are I can't do that, but technically that should be correct, no?
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The raising of the negative from the subordinate verb "know" to the main verb "do" is known by (some) linguists as (variously) not hopping, negative transportation, and neg raising, and there's at four decades' worth of discussion as to what its rules are. (This includes such things as what verbs are neg-raising verbs in various languages.) Of course, when expanding it back out you have to "lower" the negative back down to the subordinate verb again.
One randomly chosen example of the literature in the field, that gives some pointers to others: The Syntax and Semantics of Neg-Raising, with Evidence from French, Ellen F. Prince, Language, June 1976
No, there's no technical reason to assume this.
Whereas do not consists of two separate words, and only the "do" part is the "verb" for the purposes of inversion, in couldn't, shouldn't, don't, we can say that -n't is essentially a clitic, i.e. not a "word in its own right". (On phonological grounds, we might end up saying that don't is some kind of "fused" form standing in for a verb plus clitic.) A clitic is dependent upon a particular word, so that generally when that word moves, so does the clitic(s) dependent on it (like the "object pronouns" in Romance languages if you're familiar with them).
Technically one could dissolve the contraction into "Do not you know?" but this would be an awkward construction, because English favors a negative after the subject and before the verb in this interrogative form, as in "Do you not know?"
Even this sounds a bit cumbersome given the contracted alternative. It may be used for emphasis, however, as in the following sentence:
"Do you not know how to do your own laundry after all these years?"
Just to add some historical and lexicographic details. (the daily update is just too slow)
A hint is it isn’t doont, but doe-nt rhyming with toe. don't and other auxiliary-clitic combinations are no longer ‘the sum of their parts’. They have diverged some centuries ago. It’s more obvious with won't, ain't, mustn't (mussnt).
There's no synchronic way to tell why it isn’t willn’t, mustnt or doont - other than learning those forms, consciously or unconsciously.
Good question, it’s one of the points that shows grammar isn’t as simple as a ‘technical’ or ‘logical’ combination of strings of phonemes (or letters if you prefer so).
The line of thought JdeBP hints to is right, but only if you believe in (sorry, I mean it like that) underlying structures, which often require a lot of language-specific fiddling, beating the purpose of descriptions of Language-in-General.