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    In [The planar algebra of a bipartite graph, Knots in Hellas '98 (World Scientific, 2000), pp. 94–117, MR1865703], Jones found two copies of the cyclic category cΔ in the annular Temperley–Lieb category Atl. We give an abstract presentation of Atl to discuss how these two copies of cΔ 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.

    AMSC: Primary 46M15, Secondary 57M99, Secondary 18G99