Abstract
Computing polynomial form of the colored HOMFLY-PT for non-arborescent 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 ${\cal R}$-matrix. In this paper, we generalize the hypothesis to higher number of strands in the braid where commuting relations of non-neighbouring $\mathcal{R}$ matrices are also incorporated. By solving these equations, we determine the explicit form for $\mathcal{R}$-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 $SU_q(N)$ representation. Specifically, we present all the inclusive Racah matrices for representation $[2]$ and compare with the matrices obtained from eigenvalue hypothesis.
Highlights
Classification of knots is one of the most challenging research problems
We demonstrate that the eigenvalue hypothesis is still true for the case of multistrand braids
Let us denote through Ri the R matrix corresponding to the crossing between the ith and (i þ 1)th braid
Summary
Classification of knots is one of the most challenging research problems. The well-known Jones, HOMFLY-PT, and Kauffman polynomials [1,2] can distinguish many inequivalent knots but not all knots. The 6 × 6 inclusive Racah matrix was calculated in terms of the eigenvalues [29] using the Vögel universality hypothesis of Chern-Simons theory [30,31] These conjectured matrices have been independently checked. Ri , i ≥ 3 are determined by commuting with all nonneighbor R matrices [parametrized like R4 in (11)]: Ri Rj 1⁄4 Rj Ri ; jj − ij ≠ 1: In other words, according to this hypothesis, if one makes the R1 matrix diagonal with all the eigenvalues different from each other, all other matrices are uniquely defined They provide some particular representation of the braid group. We indicate the calculation of U in the three-strand braid for a specific representation These matrix elements agree with those of the eigenvalue hypothesis in Sec. III up to the sign. We will update these polynomials in our web site [34]
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