Congruent relations, cyclotomic expansion and volume conjecture for SU(n) invariants

Abstract: SU(n) invariants, including colored Jones polynomials as SU(2) invariants, are the special cases of the (reduced) colored HOMFLYPT invariants with symmetric representations. Movitated by our previous work on colored HOMFLYPT invariants, we investigate carefully the structures of SU(n) invariants. We find that SU(n) invariants possess the congruent skein relations and cyclotomic expansion. Moreover, by studying the relations between the congruent relations and limits of SU(n) invariants, we propose a volume conjecture for SU(n) invariant and we prove the conjecture for the case of figure-eight knot. This is the joint work with Qingtao Chen and Kefeng Liu.