For a twist family of knots, total knot Floer homology dimension, tau invariant and thickness grow linearly in the twist parameter, and extremal Floer homology stabilizes up to an explicit Maslov grading shift.
Abouzaid -- On the fukaya categories of higher genus surfaces , Advances in Mathematics 217 (2006), p
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Twisting, Stabilization and Bordered Floer homology
For a twist family of knots, total knot Floer homology dimension, tau invariant and thickness grow linearly in the twist parameter, and extremal Floer homology stabilizes up to an explicit Maslov grading shift.