I don't know what you know.
The sentence is ambiguous in terms of its meaning because it's ambiguous in terms of its syntax too.
It is possible that the string what you know is a fused relative here (a special kind of relative clause construction sometimes also known as a free relative). In this case the string what you know is a noun phrase. It represents an entity. These kinds of fused relatives with what can be paraphrased using the words the thing(s) that. We can paraphrase the fused relative reading of the sentence (and make it slightly clearer by adding the word same) like this:
- I don't know the same things that you know.
In this reading of the sentence we can consider the simple object of the sentence (as opposed to the full grammatical object of the sentence) as the pronoun what. This word represents the actual thing which is unfamiliar, unknown, to the speaker.
Alternatively, the string what you know could be read as an interrogative clause. In this case, if the you concerned was called Bob, for example, the sentence would mean something like:
- I don't know the answer to the question: What does Bob know?
[I used Bob in the sentence above because the deixis of you could cause further problems here]
Here the whole interrogative clause what you know represents a question, the answer to which is unknown to the speaker.
Grammars like The Cambridge Grammar of the English Language give detailed analyses of the different structures of fused relatives and interrogative clauses. The most pertinent difference is that—according to their analysis—in the fused relative reading what you know is a noun phrase (a phrase headed by a noun or pronoun, in this case the word what) , whereas what you know in the interrogative reading is an interrogative clause (and therefore ultimately headed by a verb, in this case the verb know).
There is a nice test you can do, which will tell you whether an item is a fused relative or an interrogative clause. In the interrogative clause reading you can add the word else after the what and the sentence will still make sense and still be grammatical (although it will have changed its meaning somewhat). So the following sentence can only have the interrogative reading:
- I don't know what else she knows.
The sentence above can only mean:
- I don't know the answer to the question: What else does she know?
Best Answer
Both the sentences are correct, and how you use the colon is correct in both the cases.
The colon is used to introduce something that explains or expands upon the part of the sentence it immediately follows. In other words, what comes after a colon delivers on the promise set up by what comes before it.