Thin links and Conway spheres

Abstract

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 Conway tangles, namely the Heegaard Floer invariant $\mathrm{HFT}$ and the Khovanov invariant $\widetilde{\mathrm{Kh}}$ that were developed by the authors in previous works.

arXiv comment50 pages, many colour figures. v2: Moved the classification results for Khovanov multicurves to arXiv:2202.01460

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Cited by in corpus (3)

• Khovanov homology and strong inversions
• Khovanov homology and rational unknotting
• Rasmussen invariants of Whitehead doubles and other satellites