Twist Formulas for One-Row Colored A₂ Webs and $\mathfrak {s}\mathfrak {l}_{3}$ Tails of (2, 2m)-Torus Links

Abstract

The $\mathfrak {s}\mathfrak {l}_{3}$ colored Jones polynomial $J_{\lambda }^{\mathfrak {s}\mathfrak {l}_{3}}(L)$ is obtained by coloring the link components with two-row Young diagram λ. Although it is difficult to compute $J_{\lambda }^{\mathfrak {s}\mathfrak {l}_{3}}(L)$ in general, we can calculate it by using Kuperberg’s A₂ skein relation. In this paper, we show some formulas for twisted two strands colored by one-row Young diagram in A₂ web space and compute $J_{(n,0)}^{\mathfrak {s}\mathfrak {l}_{3}}(T(2,2m))$ for an oriented (2,2m)-torus link. These explicit formulas derives the $\mathfrak {s}\mathfrak {l}_{3}$ tail of T(2,2m). They also give explicit descriptions of the $\mathfrak {s}\mathfrak {l}_{3}$ false theta series with one-row coloring because the $\mathfrak {s}\mathfrak {l}_{2}$ tail of T(2,2m) is known as the false theta series.