| Let K be a knot embedded in a Heegaard surface S for a closed orientable 3-manifold M. We call the pair (S,K) a K-splitting pair for M. Two K-splitting pairs (S,K) and (S', K') are equivalent in M if there is an ambient isotopy of M that maps (S, K) onto (S', K').;We begin this thesis by examining minimal genus K-splitting pairs in S3. Morimoto refers to this genus as the h-genus. Given a knot K in S3; we describe ways to construct K-splitting pairs using known knot invariants. These constructions provide bounds for the h-genus. We conclude this section with a brief survey of Morimoto's work concerning the additivity of the h-genus.;We begin the main part of this thesis by presenting several examples of equivalent and inequivalent K-splitting pairs. We define K-stabilization, which is similar to stabilization in the standard theory of Heegaard splittings. Motivated by the desire to characterize equivalence classes of K-splitting pairs in terms of K-stabilization, we prove the main theorem of this thesis: if (S, K) and (S', K') are K-splitting pairs of M such that K has the same surface slope with respect to both S and S', then there is a third K-splitting pair that is a K-stabilization of both. |
