The $\mathfrak{sl}_3$ colored Jones polynomials for $2$-bridge links

TL;DR

This paper derives formulas for web spaces related to Lie algebras and explicitly computes the $rak{sl}_2$ and $rak{sl}_3$ colored Jones polynomials for 2-bridge knots and links using skein theory and graphical calculus.

Contribution

It introduces new skein relations and formulas for clasped web spaces, enabling explicit calculations of $rak{sl}_2$ and $rak{sl}_3$ colored Jones polynomials for 2-bridge links.

Findings

Derived skein relations and twist formulas for web spaces.

Explicit calculations of $rak{sl}_2$ and $rak{sl}_3$ colored Jones polynomials.

Formulas generalize previous skein theory approaches.

Abstract

Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk with marked points. We derive some formulas for $A_1$ and $A_2$ clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formulas and bubble skein expansion formulas. We calculate the $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ colored Jones polynomials of $2$-bridge knots and links explicitly using twist formulas.