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 [Man07] 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.

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The first-named author was partially supported by NSF Grant DMS-1907654 and Simons Foundation Grant 524876.

The second-named author was partially supported by NSF Grant DMS-2039688 and an AMS–Simons Travel Grant.

Received 22 March 2019

Accepted 10 December 2020

Published 8 June 2022