Twist formulas for one-row colored $A_2$ webs and $\mathfrak{sl}_3$ tails of $(2,2m)$-torus links

TL;DR

This paper derives explicit formulas for the $rak{sl}_3$ colored Jones polynomial of certain torus links using $A_2$ web calculus, enabling computation of their tails and connections to false theta series.

Contribution

It introduces twist formulas for one-row colored $A_2$ webs and computes the $rak{sl}_3$ tail of $(2,2m)$-torus links, advancing the understanding of $rak{sl}_3$ link invariants.

Findings

Explicit formulas for twisted two-strand $A_2$ webs.

Computation of $rak{sl}_3$ tails for $(2,2m)$-torus links.

Connection to $rak{sl}_3$ false theta series.

Abstract

The $\mathfrak{sl}_3$ colored Jones polynomial $J_{\lambda}^{\mathfrak{sl}_3}(L)$ is obtained by coloring the link components with two-row Young diagram $\lambda$. Although it is difficult to compute $J_{\lambda}^{\mathfrak{sl}_3}(L)$ in general, we can calculate it by using Kuperberg's $A_2$ skein relation. In this paper, we show some formulas for twisted two strands colored by one-row Young diagram in $A_2$ web space and compute $J_{(n,0)}^{\mathfrak{sl}_3}(T(2,2m))$ for an oriented $(2,2m)$-torus link. These explicit formulas derives the $\mathfrak{sl}_3$ tail of $T(2,2m)$. They also give explicit descriptions of the $\mathfrak{sl}_3$ false theta series with one-row coloring because the $\mathfrak{sl}_2$ tail of $T(2,2m)$ is known as the false theta series.