Determinantal representation of colored HOMFLY for double braids

Abstract: 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 (τ)-functions. Extensions to non-rectangular representations and to more general knot families remain open problems.