We prove that the Lipshitz-Ozsváth-Thurston correspondence between extended type D structures of knot complements and $\F[U, V]/(UV)$ knot Floer complexes can be arranged so that $\iota_K$-invariant splittings of knot Floer chain complexes correspond to $\iota_{S^3 \setminus K}$-invariant splittings of bordered Floer homology of knot complements. For patterns satisfying the satellite extension property, which include cabling patterns, this provides a novel way to compute the involutive knot Floer homology of satellites from that of their companions. As a topological application, we show that our results can be applied to construct infinitely many examples of exotic pairs of contractible 4-manifolds which remain exotic after one stabilization. Along the way, we also establish first order naturality of bordered Floer homology. 

Conjecturally, a knot is slice if and only if its positive Whitehead double is slice. We consider an analogue of this conjecture for slice disks in the four-ball: two slice disks of a knot are smoothly isotopic if and only if their positive Whitehead doubles are smoothly isotopic. We provide evidence for this conjecture, using a range of techniques. More generally, we consider when isotopy obstructions persist under satellite operations. In particular, we show that obstructions coming from knot Floer homology, Seiberg-Witten theory, and Khovanov homology often behave well under satellite operations. 

We apply these strategies to give a systematic method for constructing vast numbers of exotic disks in the four-ball, including the first infinite family of pairwise exotic slice disks. These same techniques are then upgraded to produce exotic disks that remain exotic after any prescribed number of internal stabilizations. Finally, we show that the branched double covers of certain stably-exotic disks become diffeomorphic after a single external stabilization, hence stabilizing them yields exotic surfaces that have diffeomorphic branched covers.

Results of Hosokawa-Kawauchi and Baykur-Sunukjian state that homologous surfaces in a 4-manifold become isotopic after a finite number of internal stabilizations, i.e. attaching tubes to the surfaces. A natural question is how many stabilizations are needed before the surfaces become isotopic. In particular, given an exotic pair of surfaces, is a single stabilization always enough to make the pair smoothly isotopic? We answer this question by studying how the stabilization distance between surfaces with boundary changes with respect to satellite operations. Using a range of Floer theoretic techniques, we show that there are exotic disks in the four-ball which have arbitrarily large stabilization distance, giving the first examples of exotic behavior in the four-ball for which “one is not enough”.

We make use of link Floer homology to study cobordisms between links embedded in 4-dimensional ribbon homology cobordisms. Combining results of Daemi--Lidman--Vela-Vick--Wong and Zemke, we show that ribbon homology concordances induce split injections on the minus version of link Floer homology. We also make use of reduced link Floer homology to give restrictions on the number of critical points in ribbon homology concordances.