Invariant splitting principles for the Lipshitz--Ozsváth--Thurston correspondence

We prove that the Lipshitz-Ozsváth-Thurston correspondence between extended type D structures of knot complements and $\mathbb{F}[U,V]\mathrm{/}(UV)$ 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.