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

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.

29 pages, many TikZ pictures; v2: Theorem 3.17 and Theorem 5.7 corrected, Remark 5.8 and references added

References in corpus (2)

• Skein theory for SU(n)-quantum invariants
• The SL_3 colored Jones polynomial of the trefoil

Cited by in corpus (2)

• A $q$-series identity via the ${\mathfrak{sl}}_3$ colored Jones polynomials for the $(2,2m)$-torus link
• The zero stability for the one-row colored ${\mathfrak{sl}}_3$ Jones polynomial