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

Abstract

The zero stability of the colored Jones polynomial was proved by using the linear skein theory for the Kauffman bracket in Armond [Arm13]. The stability says that there exists a formal power series for a minus adequate link L such that the first n coefficients of the power series and the (n + 1)-dimensional colored Jones polynomial of L agree. In this paper, we show the zero stability of the one-row colored sl 3 Jones polynomial of a minus-adequate link by using the linear skein theory for A 2 .

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

“…The equation (3.7) is the same lemma in [Yua21b], but we prove it in a different way. We proceed by induction on n. We can see that (3.7) holds by an easy calculation for n = 2.…”

“…The following formulae help calculation of the sl 3 colored Jones polynomial. Yuasa [Yua17], [Yua18b], [Yua21b] and Kim [Kim06] gave them. We sometimes omit directions of edges of A 2 webs.…”

“…(n 1 ,n 2 ) (L; q) for torus knots. Yuasa [Yua17] [Yua18a] [Yua21a] [Yua21b]gave the explicit tails of J sl 3 (n,0) (L; q) for (2, 2m)-torus links and proved that the existence of the tails of the one-row colored sl 3 Jones polynomials for minus adequate oriented links. In this paper, we will give explicitly the one-row sl 3 colored Jones polynomials for all three-parameter family of pretzel links P(α, β, γ) except for those where α, β, γ are all odd.…”

Coloured $$\mathfrak {sl}_r$$ invariants of torus knots and characters of $${\mathcal {W}}_r$$ algebras

“…Limits of coloured invariants of knots and links have been studied in several works. We refer the readers to some important works on the existence and explicit computations of limits of g (where g is a finite-dimensional simple Lie algebra not necessarily sl 2 ) coloured invariants of various links -see [5], [6], [16], [17], [18], [22], [32], [31], [33], etc.…”

Coloured $\mathfrak{sl}_r$ invariants of torus knots and characters of $\mathcal{W}_r$ algebras

“…In the same paper [19], authors proved this conjecture for all torus knots and and all rank 2 Lie algebras. Recently, Yuasa [30] has proved 0-stability for L(nΛ 1 )-coloured sl 3 Jones polynomials for minus-adequate links using skein theory of A 2 ; see also [31] and [32].…”