An unoriented skein relation via bordered-sutured Floer homology

We show that the bordered-sutured Floer invariant of the complement of a tangle in an arbitrary 3-manifold $Y$, with minimal conditions on the bordered-sutured structure, satisfies an unoriented skein exact triangle. This generalizes a theorem by Manolescu for links in $S^3$. We give a theoretical proof of this result by adapting holomorphic polygon counts to the bordered-sutured setting, and also give a combinatorial description of all maps involved and explicitly compute them. We then show that, for $Y=S^3$, our exact triangle coincides with Manolescu’s. Finally, we provide a graded version of our result, explaining in detail the grading reduction process involved.