This thesis investigates the intersection between knot theory and the theory of 3-manifolds. 3-manifolds are well-behaved topological spaces that provide a 3-dimensional ambient space in which we study closed loops, also known as knots. Broadly speaking, the results of this thesis relate the topology of the complement of the knot in the ambient 3-manifold to various combinatorial properties of the knot.;Historically, 3-manifolds have often been studied by analyzing the surfaces they contain. Two classes of surfaces that have been closely connected to the topology and geometry of 3-manifolds are Heegaard surfaces and essential surfaces. The main result of this thesis ties together the existence of essential surfaces in the knot complement in the 3-manifold and the combinatorial properties of the knots themselves with respect to a Heegaard surface of the ambient 3-manifold. In particular, we show that if a knot has a sufficiently complicated alternating diagram with respect to a Heegaard surface, then the knot complement contains no simple essential surfaces.;In particular we show that the Euler characteristic of an essential surface in the compliment of the knot K is less than or equal to (--1/10) n where K is an n-filling alternating knot diagram.
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