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

Abstract

In this paper, a connection is established between the Jones polynomial of generalized weaving knots of type $W(3,n,m)$ and the Chebyshev polynomial of the first kind. Consequently, it is proved that the coefficients of the Jones polynomial of weaving knots are essentially the Whitney numbers of Lucas lattices. Additionally, an explicit formula for the coefficients of the Alexander polynomial of weaving knots $W(3,n)$ is introduced, and it is proven that these coefficients satisfy Fox's trapezoidal conjecture.

Keywords: Knot polynomials; Trapezoidal conjecture; Weaving knots; Whitney numbers.