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.
Categories: math.GT