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We present an algorithm for the following problem. Given a triangulated3-manifold M and a (possibly non-simple) closed curve on the boundary of M,decide whether this curve is contractible in M. Our algorithm runs in spacepolynomial in the size of the input, and (thus) in exponential time. This isthe first algorithm that is specifically designed for this problem; itconsiderably improves upon the existing bounds implicit in the literature forthe more general problem of contractibility of closed curves in a 3-manifold.The proof of the correctness of the algorithm relies on methods of 3-manifoldtopology and in particular on those used in the proof of the Loop Theorem. As abyproduct, we obtain an algorithmic version of the Loop Theorem that runs inpolynomial space, and (thus) in exponential time.

Deciding contractibility of a non-simple curve on the boundary of a 3-manifold: A computational Loop Theorem