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 closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving mathrm{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{mathrm{mathcal{R}}} -matrices can be written in terms of infinite family of Laurent polynomials {mathcal{V}}_{n,t}left[qright] whose absolute coefficients has interesting relation to the Fibonacci numbers {mathrm{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 first kind.
Highlights
Derived from the new knot invariants is related to natural differential geometric invariants becomes another natural problem
We confine to a hybrid generalization of W (3, n) which we denote as W 3(m, n) and obtain closed form expression for HOMFLYPT using the Reshitikhin and Turaev method involving R-matrices
We use the approach of Reshitikhin and Turaev to evaluate the colored polynomials for knots and obtained the closed form expression for HOMFLY-PT polynomial for hybrid weaving knots
Summary
Recall Alexander theorem which states that any knot or link can be viewed as closure of m-strand braid. The knot invariants can be constructed from the braid group Bm representations. According to Reshetikhin-Turaev approach [16, 17] the quantum group invariant, known as [r]-colored HOMFLY polynomial of the knot K denoted by H[Kr] is defined as follows: H[Kr] = qtrV1⊗···⊗Vm (π(αK)) ,. There is a modified RT-approach [19,20,21] where the braiding generators can be written in a block structure form. This methodology gives a better control and simplify the computation of knot invariants. We will present the details of this modified RT method
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