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

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 ${\mathfrak{sl}}_3$.

28 pages, many TikZ pictures; v2: The proof of the zero-stability is restricted for B-adequate links "with antiparallel twist regions'' in this version. The poof for links without the restriction is omitted, and we will discuss it in the forthcoming paper

References in corpus (1)

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