A self-pairing theorem for tangle Floer homology

Abstract

We show that for a tangle $T$ with $-{\partial }^{0}T\cong {\partial }^{1}T$ the Hochschild homology of the tangle Floer homology $\widetilde{CT}\left(T\right)$ is equivalent to the link Floer homology of the closure $T^{\prime }=T∕\left(-{\partial }^{0}T\sim {\partial }^{1}T\right)$ 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.