Invariant splitting principles for the Lipshitz–Ozsváth–Thurston correspondence

Abstract We prove that the Lipshitz–Ozsváth–Thurston correspondence between extended type D structures of knot complements and knot Floer complexes can be arranged so that ‐invariant splittings of knot Floer chain complexes correspond to ‐invariant splittings of bordered Floer homology of knot complements. 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.