When does Dehn surgery along a knot give an L-space? More generally, when does splicing two knot complements give an L-space? Hanselman, Rasmussen and Watson gave very compelling answers to these questions using their technology of immersed curves for three-manifolds with torus boundary. Similar invariants have been developed for four-ended tangles. We use those invariants to study various notions of thinness in both Heegaard Floer and Khovanov homology from the perspective of tangle decompositions along Conway spheres. Interestingly, our results bear strong resemblance to the aforementioned results about L-spaces. Also, we observe strong similarities between Heegaard Floer and Khovanov homology that lead us to ask: What is a thin link? This is joint work with Artem Kotelskiy and Liam Watson.