Counting in Calabi--Yau categories, with applications to Hall algebras and knot polynomials

We show that homotopy cardinality – a priori ill-defined for many dg-categories, including all periodic ones – has a reasonable definition for even-dimensional Calabi–Yau (evenCY) categories and their relative generalizations (under appropriate finiteness conditions). As a first application we solve the problem of defining an intrinsic Hall algebra for degreewise finite pre-triangulated dg-categories in the case of oddCY categories. We compare this definition with Toën’s derived Hall algebras (in case they are well-defined) and with other approaches based on extended Hall algebras and central reduction, including a construction of Hall algebras associated with Calabi–Yau triples of triangulated categories. For a category equivalent to the root category of a 1CY abelian category $\mathcal{A}$, the algebra is shown to be isomorphic to the Drinfeld double of the twisted Ringel–Hall algebra of $\mathcal{A}$, thus resolving in the Calabi–Yau case the long-standing problem of realizing the latter as a Hall algebra intrinsically defined for such a triangulated category. Our second application is the proof of a conjecture of Ng–Rutherford–Shende–Sivek, which provides an intrinsic formula for the ruling polynomial of a Legendrian knot $L$, and its generalization to Legendrian tangles, in terms of the augmentation category of $L$.