Recently the radiative $B$ decay to the strange axial-vector mesons, $B¥to K_1(1270) ¥gamma$, has been observed with rather large rate. This process is particularly interesting as the subsequent $K_1$ decay into its three body final state allows to determine the polarization of the gamma, which is mostly left- (right-)handed for $¥overline{B} (B)$ in the SM while various new physics models predict additional right- (left-)handed components. A new method is proposed to determine the polarization, exploiting the full Dalitz plot distribution, which seems to reduce strongly the statistical errors. In order to obtain a theoretical prediction for this polarization measurement, it is necessary to understand the hadronic $K_1 ¥to K ¥pi ¥pi$ decay channel and its uncertainties. The strong decays of the $K_1$ mesons, namely the partial wave amplitudes as well as their relative phases, are revisited in the framework of the $^3P_0$ quark-pair-creation model. Then, the result on the sensitivity of the $B¥to K_1(1270) ¥gamma$ process to the photon polarization is presented.