The one-row colored $\mathfrak{sl}_{3}$ Jones polynomials for pretzel links

TL;DR

This paper computes the one-row $ ext{sl}_3$ colored Jones polynomials for certain pretzel links using skein theory and demonstrates the existence of polynomial tails for specific alternating pretzel knots.

Contribution

It provides explicit formulas for the one-row $ ext{sl}_3$ Jones polynomials of pretzel links and proves tail existence for a family of alternating pretzel knots.

Findings

Explicit formulas for $J_{(n, 0)}^{ ext{sl}_3}$ for pretzel links

Calculation method using Kuperberg's skein theory

Proof of tail existence for certain alternating pretzel knots

Abstract

The colored $\mathfrak{sl}_{3}$ Jones polynomial $J_{(n_{1}, n_{2})}^{\mathfrak{sl}_{3}}(L;q)$ are given by a link and an $(n_{1}, n_{2})$-irreducible representation of $\mathfrak{sl}_{3}$. In general, it is hard to calculate $J_{(n_{1}, n_{2})}^{\mathfrak{sl}_{3}}(L;q)$ for an oriented link $L$. However, we calculate the one-row $\mathfrak{sl}_{3}$ colored Jones polynomials $J_{(n, 0)}^{\mathfrak{sl}_{3}}(P(\alpha,\beta,\gamma);q)$ for three-parameter families of oriented pretzel links $P(\alpha,\beta,\gamma)$ by using Kuperberg's linear skein theory by setting $n_{2}=0$. Furthermore, we show the existence of the tails of $J_{(n, 0)}^{\mathfrak{sl}_{3}}(P(2\alpha +1, 2\beta+1,2\gamma);q)$ for the alternating pretzel knots $P(2\alpha +1, 2\beta+1,2\gamma)$.