  Weaving knots $W(p, n)$ of type $(p, n)$ denote an infinite family of
hyperbolic knots which have not been addressed by the knot theorists as yet.
Unlike the well-known $(p,n)$ torus knots, we do not have a closed-form
expression for HOMFLY-PT and the colored HOMFLY-PT for $W(p,n)$. In this paper,
we confine to a hybrid generalization of $W(3,n)$ which we denote as
$\hat{W}_3(m,n)$ and obtain a closed-form expression for HOMFLY-PT using the
Reshitikhin and Turaev method involving $\mathcal R$-matrices. Further, we also
compute $[r]$-colored HOMFLY-PT for $W(3,n)$. Surprisingly, we observe that
trace of the product of two dimensional $\hat{\mathcal{R}}$-matrices can be
written in terms of an infinite family of Laurent polynomials
$\mathcal{V}_{n,t}[q]$ whose absolute coefficients has an interesting relation
to the Fibonacci numbers $\mathcal{F}_{n}$. We also computed reformulated
invariants and the BPS integers in the context of topological strings. From our
analysis, we propose that certain refined BPS integers for weaving knot
$W(3,n)$ can be explicitly derived from the coefficients of Chebyshev
polynomials of the first kind.
