Generalized knots–quivers correspondence

Abstract

We propose a generalized version of knots–quivers correspondence, where the quiver series variables specialize to arbitrary powers of the knot HOMFLY-PT polynomial series variable. We explicitly compute quivers for large classes of knots, as well as many homologically thick 9- and 10-crossings knots, including the ones with the super-exponential growth property of colored HOMFLY-PT polynomials. In addition, we propose a new, compact, quiver-like form for the colored HOMFLY-PT polynomials, where the structure of colored differentials is manifest. In particular, this form partially explains the non-uniqueness of quivers corresponding to a given knot via knots–quivers correspondence.