Thin links and Conway spheres.

When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal $\delta$-grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for four-ended tangles, namely the Heegaard Floer invariant $\operatorname{HFT}$ and the Khovanov invariant $\operatorname{\widetilde{Kh}}$ that were developed by the authors in previous works. Applying ideas from homological mirror symmetry, we show that $\operatorname{\widetilde{Kh}}$ is subject to the same strong geography restrictions that were already known for $\operatorname{HFT}$.