Thin links and Conway spheres

When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal δ -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.