Zero-knowledge proof system which is not proof of knowledge? I have learned that a zero-knowledge (ZK) proof system can be constructed by making use of the three-color problem. In this particular case, the proof system also happens to be proof of knowledge (POK), i.e. any prover necessarily needs to know a witness in order to convince the verifier.

Are there ZK proof systems which are not POK?
There are certainly ZK proof systems which are not known to be POK, and for which no knowledge extractor is known. For example, take the Goldreich-Kahan 4-round ZK proof system. However, do we know of a non-trivial proof system that is provably not a proof of knowledge? Not that I know of.