A self-pairing theorem for tangle Floer homology

TL;DR

This paper establishes a self-pairing theorem linking Hochschild homology of tangle Floer homology to link Floer homology of the closure, and proves the braid group's faithful action on tangle Floer homology.

Contribution

It introduces a self-pairing theorem for tangle Floer homology and demonstrates the faithfulness of the braid group action on this homology.

Findings

Hochschild homology of tangle Floer homology equals link Floer homology of the closure.

Braid group action on tangle Floer homology is faithful.

Provides a new perspective on the relationship between tangle and link Floer homologies.

Abstract

We show that for a tangle $T$ with $-\partial^0T \cong \partial^1 T$ the Hochschild homology of the tangle Floer homology $\widetilde{\mathit{CT}}(T)$ 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.