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When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal -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 Conway tangles, namely the Heegaard Floer invariant HFT and the Khovanov invariant {Kh} that were developed by the authors in previous works.
Kotelskiy et al. (Mon,) studied this question.