$C_n$-moves and the difference of Jones polynomials for links

The Jones polynomial $V_L(t)$ for an oriented link $L$ is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer $n\ge 3$, we show that: (1) the difference of Jones polynomials for two oriented links which are $C_n$-equivalent is divisible by ${(t-1)}^n(t^2+t+1)(t^2+1)$, and (2) there exists a pair of two oriented knots which are $C_n$-equivalent such that the difference of the Jones polynomials for them equals ${(t-1)}^n(t^2+t+1)(t^2+1)$.