You answered your own question.  Do not submit to any type of "lie detector" test.  It's their burden to find out who did it using legitimate forensic science, such as fingerprints.  Some might tell you to get a lawyer, but it might be early for that.  Just politely decline the test.  Good luck.

down and out,

Thank you for your kind words regarding AntiPolygraph.org. This website was created for the benefit of people who find themselves in situations such as yours, as well as those who have already fallen victim to the quackery of the lie detector.

I didn't reply earlier because I think mustbaliar's reply was very much to the point. But it is perhaps worth noting in addition that when the police told you that fingerprints were found on the jewelry box, it may have been a simple interrogator's ruse. Here's how it works: the interrogator falsely claims that fingerprints were found at the scene of the crime and asks the suspect if he/she would be willing to provide his/her prints. In fact, no fingerprints were found, and the real purpose is to see if the suspect will willingly provide prints. If the suspect refuses or hesitates, guilt is inferred. Conversely, if the suspect readily agrees to provide prints, innocence might be inferred. This is actually a pretty clever interrogational ruse that I do not necessarily find objectionable, though investigators should bear in mind that it is far from infallible. An innocent person might still refuse to submit to voluntary fingerprinting while a guilty person might readily agree to it.

The fact that the police tape-recorded their interview of you is a good thing, as it creates an objective record of what was said (and not said) during the interview/interrogation. Some law enforcement agencies, such as the FBI, cynically refuse to tape record interrogations precisely to prevent any such objective record from being created. Then if there's any dispute about what was said, it's the word of a law enforcement officer against that of an accused criminal. The FBI seems to like it that way. But it doesn't serve the interest of equal justice under the law.

As for voice stress analysis, it's sheer quackery. The "National Institute of Truth Verification", manufacturer of the most widely used voice stress analyzer, has admitted in court that their device is not capable of lie detection. Voice stress analysis is a complete and utter fraud that has regrettably been embraced by numerous law enforcement agencies. As with the polygraph, many detectives are evidently stupid or credulous enough to actually believe in it. You are wise to refuse to submit to this pseudoscientific nonsense.