A self-pairing theorem for tangle Floer homology

We show that for a tangle $T$ with $-{\mathrm{\partial }}^{0}T\cong {\mathrm{\partial }}^{1}T$ the Hochschild homology of the tangle Floer homology $\widetilde{CT}(T)$ is equivalent to the link Floer homology of the closure $T’ = T/(-\partial^0T \sim \partial^1 T)$ of the tangle, linked with the tangle axis. In addition, we show that the action of the braid group on tangle Floer homology is faithful.