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It is often the case that the study of knots leads to questions in 3-manifold theory and vice-versa. One of the ways that this can happen is to ask a question about a knot and then turn to its complement in the 3-sphere for the answer. The question I have is this: Is there some combinatorial condition for a knot's complement that excludes certain essential surfaces from appearing inside it? In particular, can we exhibit a combinatorial condition for the knot's complement to be atoroidal?
My aim is to set up the background for this question (including results in thin position and reducibility of Heegaard splittings a la Casson and Gordon), explain some of the results in certain cases, and give an idea of what difficulties arise in general.