Volume conjecture for SU(n)-invariants

This paper discuss an intrinsic relation among congruent relations [1], cyclotomic expansion and Volume Conjecture for SU(n) invariants. Motivated by the congruent relations for SU(n) invariants obtained in our previous work [1], 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.