  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.
