Determinantal representation of colored HOMFLY for double braids

Starting from the known differential expansions, we express the HOMFLY-PT polynomials of twist and antiparallel double-braid knots in rectangular representations $[r^s]$ as determinants of $r\times r$ and $s\times s$ matrices. We first give an elementary derivation for the figure-eight knot and then re-sum the KNTZ formulas for twist and double-braid evolution. The resulting matrices are constructed from single-hook evolution coefficients and explicit representation-dependent weights. These formulas provide a starting point for investigating possible extensions of knot polynomials to KP/Toda $\tau$-functions. Extensions to non-rectangular representations and to more general knot families remain open problems.