A self-pairing theorem for tangle Floer homology (Q312392)

Knot Floer homology is an invariant introduced by Ozsváth-Szabó and, independently, Rasmussen; further refinements extended the definition to links and gave a fully combinatorial construction. The variant considered here is usually called \(\widetilde{\mathrm{HFK}}\); this is not an invariant of the isotopy type of the link, but it additionally depends on the number of basepoints in a Heegaard diagram.NEWLINENEWLINETangle Floer homology is an invariant associated to tangles introduced by the two authors, denoted by \(\widetilde{\mathrm{CT}}\); it combines features of the combinatorial construction of knot Floer homology and of bordered Floer homology. As for \(\widetilde{\mathrm{HFK}}\), this, too, depends on the number of basepoints in a Heegaard diagram.NEWLINENEWLINEWhen a tangle \(\mathcal T\) in a 3-manifold \(Y\) with two \(S^2\)-boundary components is \textit{strongly marked}, it naturally gives rise to a link \(T_0\) in a closed 3-manifold \(Y_0\) by a construction that is similar to braid closure. The main theorem of the paper relates the tangle Floer homology of \(\mathcal T\) to the link Floer homology of \(T_0\) via Hochschild homology. Namely, the authors prove that NEWLINE\[NEWLINE HH(\widetilde{\mathrm{CT}}(\mathcal{T})) \cong \widetilde{\mathrm{HFK}}(T_0). NEWLINE\]NEWLINE As an application, they obtain that tangle Floer homology restricts to a faithful action of the braid group on the homotopy category of right type \(A\) modules over the differential algebra associated to an \(n\)-pointed sphere.