Volume conjecture for $SU(n)$-invariants

Abstract

This paper discuss an intrinsic relation among congruent relations \cite{CLPZ}, cyclotomic expansion and Volume Conjecture for $SU(n)$ invariants. Motivated by the congruent relations for $SU(n)$ invariants obtained in our previous work \cite{CLPZ}, we study certain limits of the $SU(n)$ invariants at various roots of unit. First, we prove a new symmetry property for the $SU(n)$ invariants by using a symmetry of colored HOMFLYPT invariants. Then we propose some conjectural formulas including the cyclotomic expansion conjecture and volume conjecture for $SU(n)$ invariants (specialization of colored HOMFLYPT invariants). We also give the proofs of these conjectural formulas for the case of figure-eight knot.

17 pages

References in corpus (3)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Congruent skein relations for colored HOMFLY-PT invariants and colored Jones polynomials
• On the formulae for the colored HOMFLY polynomials

Cited by in corpus (4)

• Congruent skein relations for colored HOMFLY-PT invariants and colored Jones polynomials
• Evolution properties of the knot's defect
• Asymptotic Behavior of Colored HOMFLY Polynomial of Figure Eight Knot
• Asymptotic Behavior of Colored Jones polynomial and Turaev-Viro Invariant of figure eight knot