Alexander and Jones Polynomials of weaving 3-braid links and Whitney rank polynomials of Lucas lattice

Abstract

We establish a relationship between the Jones polynomial of generalized weaving knots of type $W(3,n,m)$ and the Chebyshev polynomial of the first kind. Consequently, we prove that the coefficients of the Jones polynomial of weaving knots are basically the Whitney numbers of Lucas lattices. Furthermore, we give an explicit formula for the Alexander polynomial of weaving knots $W(3,n)$ and we prove that it satisfies Fox's trapezoidal conjecture.