Year: 2018
Type: article
Abstract: 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 $\mathcal{R}$ matrix. In this paper, we generalize the hypothesis to higher number of strands in the braid where commuting relations of non-neighboring $\mathcal{R}$ 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 $S{U}_{q}(N)$ representation. Specifically, we present all the inclusive Racah matrices for representation [2] and compare with the matrices obtained from eigenvalue hypothesis.
Authors: Saswati Dhara, A. Mironov, A. Morozov, An. Morozov, P. Ramadevi, Vivek Kumar Singh, A. Sleptsov
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W2774314549