A stability conjecture for the colored Jones polynomial

Abstract

We formulate a stability conjecture for the coefficients of the colored Jones polynomial of a knot, colored by irreducible representations in a fixed ray of a simple Lie algebra, and verify it for all torus knots and all simple Lie algebras of rank $2$. Our conjecture is motivated by a structure theorem for the degree and the coefficients of a $q$-holonomic sequence of polynomials given in [Ga2] and by a stability theorem of the colored Jones polynomial of an alternating knot given in \cite{GL2}. We illustrate our results with sample computations.

32 pages, 4 figures

References in corpus (1)

• Nahm sums, stability and the colored Jones polynomial

Cited by in corpus (4)

• A $q$-series identity via the ${\mathfrak{sl}}_3$ colored Jones polynomials for the $(2,2m)$-torus link
• Higher depth quantum modular forms, multiple Eichler integrals, and ${\mathfrak{sl}}_3$ false theta functions
• Torus Knots and the Chern-Simons path integral: a rigorous treatment
• On the degree of the colored Jones polynomial