In this dissertation, we present the culmination of two projects, after an overview of theprimary tool involved in the research, Heegaard Floer theory. In this overview, we discuss the origins of Heegaard Floer homology, an invariant associated to a Spinc 3-manifold, as well as its flavors. We then present multiple flavors of knot Floer homology, a refinement of that theory. The first project is a structural theorem for a family of knot invariants due to Dowlin. L-space knots are knots which admit surgeries that have simple Heegaard Floer homology and thin knots are ones whose knot Floer homology is concentrated in a single δ-grading. Each class of knots has well known knot Floer complexes. As such, we show that for L-space knots and thin knots, the theories that Dowlin constructed are a change of coefficients from an older theory, the minus flavor of knot Floer homology. Many supporting examples are shown in its final section. The proof uses a popular cancellation lemma for chain complexes with the special shapes involved. The second project is a collaboration with McConkey, St. Clair, and Zhang. In this dissertation, we show that the Whitehead double of the dual knot to 1/n surgery on the knot 61 in the 3-sphere is deeply slice in a contractible 4-manifold. That is, it bounds a smoothly embedded disc in the manifold, but not in a collar neighborhood of its boundary, the surgered manifold. This is partial progress in answering one of the Kirby questions regarding nullhomotopic deeply slice knots, as referenced in earlier work of Klug and Ruppik. To prove our theorem, we make use of the immersed curves perspective of bordered Floer homology and knot Floer homology, which we introduce in previous sections.
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