Behavior of knot Floer homology under conway and genus two mutation

In this dissertation we prove that if an n-stranded pretzel knot K has an essential Conway sphere, then there exists an Alexander grading s such that the rank of knot Floer homology in this grading, [mathematical equation], is at least two. As a consequence, we are able to easily classify pretzel knots admitting L-space surgeries. We conjecture that this phenomenon occurs more generally for any knot in S³ with an essential Conway sphere. We also exhibit an infinite family of knots, each of which admits a nontrivial genus two mutant which shares the same total dimension of knot Floer homology, while being distinguished by knot Floer homology as a bigraded invariant. Additionally, the genus two mutation interchanges the [mathematical symbol]-graded knot Floer homology groups in [mathematical symbol]-gradings k and -k. This infinite family of examples supports a second conjecture, namely that the total rank of knot Floer homology is invariant under genus two mutation.