## Abstract

Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a pair: bicategory Open image in new window and endobifunctor Open image in new window . For a graded linear bicategory and a fixed invertible parameter *q*, we quantize this theory by using the endofunctor \(\Sigma _q\) such that \(\Sigma _q \alpha :=q^{-\deg \alpha }\Sigma \alpha \) for any 2-morphism \(\alpha \) and coincides with \(\Sigma \) otherwise. Applying the quantized trace to the bicategory of Chen–Khovanov bimodules we get a new triply graded link homology theory called *quantum annular link homology*. If \(q=1\) we reproduce Asaeda–Przytycki–Sikora homology for links in a thickened annulus. We prove that our homology carries an action of Open image in new window , which intertwines the action of cobordisms. In particular, the quantum annular homology of an *n*-cable admits an action of the braid group, which commutes with the quantum group action and factors through the Jones skein relation. This produces a nontrivial invariant for surfaces knotted in four dimensions. Moreover, a direct computation for torus links shows that the rank of quantum annular homology groups depend on the quantum parameter *q*.

## Notes

### Acknowledgements

The authors are grateful to Adrien Brochier, Matthew Hogancamp, Mikhail Khovanov, Slava Krushkal, Aaron Lauda, David Rose, and Paul Wedrich for stimulating discussions. During an early stage of the research Robert Lipshitz suggested to look on higher Hochschild homology of the arc algebras and Ben Webster pointed a connection between Hochschild homology and the global dimension. The first two authors are supported by the NCCR SwissMAP founded by the Swiss National Science Foundation. The third author was supported by the NSF Grant DMS-1111680.

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