Heegaard Floer multicurves of double tangles

Abstract

We describe a simple formula for computing the Heegaard Floer multicurve invariant of double tangles from the Heegaard Floer multicurve invariant of knot complements. A comparison with a similar multicurve invariant for Conway tangles in the setting of Khovanov homology confirms that knot Floer homology and Khovanov homology behave very differently under satellite operations, echoing recent observations from arXiv:2208.13612. We also obtain a new characterisation of L-space knots in terms of Heegaard Floer A-link satellites. Along the way, we find the first example of a satellite knot whose knot Floer homology is thin.

31 pages, colour figures; written for the proceedings of "Frontiers in Geometry and Topology", a conference in honour of Tom Mrowka's 60th birthday; v2: add applications of the main theorem about thinness and A-links, including the first example of a Heegaard Floer thin satellite knot; expand discussion of torsion orders; add script that implements the main theorem