Heegaard Floer multicurves of double tangles

TL;DR

This paper introduces a formula for computing Heegaard Floer multicurve invariants of double tangles, compares it with Khovanov homology invariants, and characterizes L-space knots, revealing differences in satellite operations.

Contribution

It provides a new simple formula for Heegaard Floer multicurve invariants of double tangles and offers a novel characterization of L-space knots.

Findings

Heegaard Floer multicurve invariant for double tangles can be computed from knot complements.

Knot Floer homology and Khovanov homology behave differently under satellite operations.

Identified the first satellite knot with thin knot Floer homology.

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.