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Ramadevi, Rama Mishra, Vivek Kumar Singh","submitted_at":"2021-03-18T13:00:19Z","abstract_excerpt":"Weaving knots $W(p, n)$ of type $(p, n)$ denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well-known $(p,n)$ torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for $W(p,n)$. In this paper, we confine to a hybrid generalization of $W(3,n)$ which we denote as $\\hat{W}_3(m,n)$ and obtain a closed-form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $\\mathcal R$-matrices. Further, we also compute $[r]$-colored HOMFLY-PT for $W(3,n)$. 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