The Existence of a Meridional Curve in Closed Incompressible Surfaces in Fully Alternating Link Complements

Menasco showed that a closed incompressible surface in the complement of a non-split prime alternating link in S³ contains a circle isotopic in the link complement to a meridian of the links. Based on this result, he was able to argue the hyperbolicity of non-split prime alternating links in S³. Adams et al. showed that if F ⊂ S × I L is an essential torus, then F contains a circle which is isotopic in S × I \ L to a meridian of L. The author generalizes his result as follows: Let S be a closed orientable surface, L be a fully alternating link in S × I. If F ⊂ S × I \ L is a closed essential surface, then F contains a circle which is isotopic in S × I \ L to a meridian of L.