The zero stability for the one-row colored $\mathfrak{sl}_3$ Jones polynomial

TL;DR

This paper proves the zero stability of one-row colored $ ext{sl}_3$ Jones polynomials for $B$-adequate links with anti-parallel twist regions, extending stability results beyond $ ext{sl}_2$ using skein theory.

Contribution

It establishes zero stability for $ ext{sl}_3$ Jones polynomials of $B$-adequate links, employing Kuperberg's $ ext{sl}_3$-webs, advancing understanding of quantum invariants.

Findings

Proves zero stability for $ ext{sl}_3$ Jones polynomials.

Demonstrates existence of related $q$-series from quantum invariants.

Extends stability results from $ ext{sl}_2$ to $ ext{sl}_3$.

Abstract

The stability of coefficients of colored ($\mathfrak{sl}_2$-) Jones polynomials $\{J_{K,n}^{\mathfrak{sl}_2}(q)\}_n$ was discovered by Dasbach and Lin. This stability is now called the zero-stability of $J_{K,n}^{\mathfrak{sl}_2}(q)$. Armond showed zero stability for a $B$-adequate link by using the linear skein theory based on the Kauffman bracket. In this paper, we prove the zero stability of one-row colored $\mathfrak{sl}_{3}$-Jones polynomials $\{J_{K,n}^{\mathfrak{sl}_3}(q)\}_n$ for $B$-adequate links $L$ with anti-parallel twist regions by using the linear skein theory based on Kuperberg's $\mathfrak{sl}_3$-webs. It implies the existence of many $q$-series obtained from a quantum invariant associated with $\mathfrak{sl}_3$.