  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.
