A Cyclic Approach to the Annular Temperley-Lieb Category

In 2000, Jones found two copies of the cyclic category in the annular Temperley-Lieb category ATL. We give an abstract presentation of ATL to discuss how these two copies of the cyclic category generate ATL together with the coupling constants and the coupling relations. We then discuss modules over the annular category and homologies of such modules, the latter of which arises from the cyclic viewpoint.