Invariant splitting principles for the Lipshitz--Ozsv\'ath--Thurston correspondence

We prove that the Lipshitz-Ozsv\'ath-Thurston correspondence between extended type D structures of knot complements and FU, V/ (UV) knot Floer complexes can be arranged so that K-invariant splittings of knot Floer chain complexes correspond to ₒ℃ ₊-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.