We consider the kink modes in a zero-beta cylindrical compressible thin magnetic flux tube modelled as a twisted core surrounded by a magnetically twisted annulus, both embedded in a straight ambient external field. The dispersion relation is derived and solved numerically to obtain the frequencies of the kink waves. The main result is that the twisted magnetic annulus does affect the period ratio P1/P2 of the kink modes. For the kink modes, the magnetic twist in the annulus region can achieve deviations from P1/P2 = 2 of the same order of magnitude as in the observations.
