Menasco showed that a non-split, prime, alternating link that is not a 2-braid is hyperbolic in S 3 . We prove a similar result for links in closed thickened surfaces S × I. We define a link to be fully alternating if it has an alternating projection from S × I to S where the interior of every complementary region is an open disk. We show that a prime, fully alternating link in S × I is hyperbolic. Similar to Menasco, we also give an easy way to determine primeness in S ×I. A fully alternating link is prime in S ×I if and only if it is "obviously prime". Furthermore, we extend our result to show that a prime link with fully alternating projection to an essential surface embedded in an orientable, hyperbolic 3-manifold has a hyperbolic complement.
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