Colored HOMFLY-PT for hybrid weaving knot ${\hat{W}}_3(m,n)$

Abstract

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 $W(3,n)$. Surprisingly, we observe that trace of the product of two dimensional $\hat{\mathcal{R}}$-matrices can be written in terms of an infinite family of Laurent polynomials ${\mathcal{V}}_{n,t}[q]$ whose absolute coefficients has an interesting relation to the Fibonacci numbers ${\mathcal{F}}_n$. We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot $W(3,n)$ can be explicitly derived from the coefficients of Chebyshev polynomials of the first kind.

32 pages, typos corrected

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