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tokens: form from that obtained for the fundamental frequencies of crustal torsional oscillations in the phases of spherical and cylindrical nuclei (Sotani et al. 2018 ). We also remark that the fundamental frequency in the tube and bubble phases can be smaller than that in the phases of spherical and cylindrical nuclei and that the fundamental frequencies with general values of l can also be well fitted in the same functional form: with a different set of the adjustable parameters , , and . Hereafter, we will thus attempt to identify the observed QPO frequencies using the fitting formula ( 10 ) for the tube–bubble layer, in addition to the formula given in Sotani et al. ( 2018 ) for the sphere–cylinder layer. Figure 2. (Colour online) Fundamental frequencies of the l = 2 torsional oscillations in the tube and bubble phases, as obtained for the neutron star models with M = 1.4M ⊙ and R = 12 km as well as with various sets of L and K 0 . Here, the left-hand and right-hand panels correspond to the results for the maximum and minimum enthalpies that participate in the oscillations, i.e. and 0, respectively (see the text for details). The solid line denotes the fitting given by equation ( 9 ). Note that the obtained fundamental frequencies in the tube–bubble layer are generally lower than those obtained in our earlier analysis by assuming that only the bubble phase is oscillating (Sotani et al. 2017a ). This tendency is more significant for larger values of L . This is partly because for larger L , the bubble phase is less likely to appear, as shown in Table 1 , and partly because the shear modulus and hence the shear velocity is relatively small in the tube phase, as shown in Fig. 1 . As we will see later, therefore the 26 Hz QPO observed from SGR 1806-20 is identified as the l = 4 fundamental torsional oscillation in the tube–bubble layer, in contrast to our earlier analysis (Sotani et al. 2017a ) in which it was identified as the l = 2 fundamental torsional oscillation. Up to now, we have already done many trials to identify the QPOs observed in SGR 1806-20 and SGR 1900+14 as the crustal torsional oscillations. As long as we adopt the QPO frequencies derived in the conventional non-Bayesian analyses of RXTE data for the X-ray afterglows of the giant flares and the recurrent X-ray outbursts, such identification has worked out relatively well (Sotani et al. 2012 , 2013a , b , 2016 , 2017a , 2018 ). In fact, the observed QPOs, except for the 26 Hz QPO in SGR 1806-20, can be identified as the torsional oscillations inside the phases of spherical and cylindrical nuclei in such a way that the QPOs of frequencies lower than ∼200 Hz correspond to the fundamental oscillations with various values of l, while the higher QPOs observed in SGR 1806-20 correspond to the overtones (Sotani et al. 2018 ). In this case, since it is still uncertain how much fraction of dripped neutrons in the phase of cylindrical nuclei would be locked to the motion of protons in the