The United States Patent and Trademark Office has rejected a request from Nicole Polizzi, a k a Snooki from MTV’s “Jersey Shore,” to trademark her nickname. According to a Trademark Office document published on the Web site TheSmokingGun.com, Snooki cannot be held as a trademark by Ms. Polizzi because of the “likelihood of confusion” with a 2004 trademark belonging to Brian J. Publishing, publisher of the children’s book “Adventures of Snooky: Under the Sea,” by Karen Monaghan-Arnone and Maureen Monaghan-Faber, about a cat lost at sea. Ms. Polizzi filed her request on Feb. 7 for the purpose of “entertainment in the nature of personal appearances by a television personality,” according to the Trademark Office documents. Ms. Polizzi may appeal the decision if she chooses.