Colored HOMFLY-PT for hybrid weaving knot $\hat{W}_{3}(m,n)$

TL;DR

This paper derives a closed-form expression for the HOMFLY-PT polynomial of a hybrid family of weaving knots, revealing connections to Fibonacci numbers and topological string invariants.

Contribution

It introduces a new hybrid generalization of weaving knots and provides explicit formulas for their HOMFLY-PT and colored HOMFLY-PT polynomials using R-matrix methods.

Findings

Closed-form HOMFLY-PT for $ ilde{W}_3(m,n)$ derived.

Trace of R-matrix products linked to Fibonacci-related Laurent polynomials.

Reformulated invariants and BPS integers computed, with relations to Chebyshev polynomials.

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.