Even if Knox fails in all her appeals (which isn't going to happen, but let's hypothesise

), the US authorities will absolutely make no public complaints. There may well have been private conversations between (e.g.) the US Ambassador to Rome and a senior judicial figure in Italy, but no coercion or even suggestion - that would be entirely improper, and way outside the US's remit. The conversation (if there even was one) would have gone no further than a reminder that the outcome of the appeal process is being closely followed in the US and UK. There will have been no threat of trade restrictions or reduced military cooperation (or anything else) - that just doesn't happen in the case of the prosecution of a private citizen for a crime which is not state-related. And it won't happen here, no matter what happens in the appeals.