Eigenvalue hypothesis for multistrand braids

Computing polynomial form of the colored HOMFLY-PT for nonarborescent knots obtained from three or more strand braids is still an open problem. One of the efficient methods suggested for the three-strand braids relies on the eigenvalue hypothesis which uses the Yang-Baxter equation to express the answer through the eigenvalues of the R matrix. In this paper, we generalize the hypothesis to higher number of strands in the braid where commuting relations of non-neighboring R matrices are also incorporated. By solving these equations, we determine the explicit form for R matrices and the inclusive Racah matrices in terms of braiding eigenvalues (for matrices of size up to 6 by 6). For comparison, we briefly discuss the highest weight method for four-strand braids carrying fundamental and symmetric rank two S U_q(N ) representation. Specifically, we present all the inclusive Racah matrices for representation [2] and compare with the matrices obtained from eigenvalue hypothesis.