$C_{n}$-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 $nge 3$, we show that: (1) the difference of Jones polynomials for two oriented links which are $C_{n}$-equivalent is divisible by $left(t-1right)^{n}left(t^{2}+t+1right)left(t^{2}+1right)$, 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 $left(t-1right)^{n}left(t^{2}+t+1right)left(t^{2}+1right)$.