Cluster Algebras and the HOMFLY Polynomial

Recently, it has been shown that the Jones polynomial, in [LS19], and the Alexander polynomial, in [NT18], of rational knots can be obtained by specializing $F$-polynomials of cluster variables. At the core of both results are continued fractions, which parameterize rational knots and are used to obtain cluster variables, by way of snake graphs in the case of [LS19], or ancestral triangles in the case of [NT18]. In this paper, we use path posets, another structure parameterized by continued fractions, to directly generalize [LS19]'s construction to a specialization yielding the HOMFLY polynomial, which generalizes both the Jones and Alexander polynomials.