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States generated in the $K$-multi-$ρ$ interactions

In the present work we use three-body interaction formalism to investigate the $K$-multi-$ρ$ interactions. First, we reproduced the resonances $f_2(1270)$ and $K_1(1270)$ in the $ρρ$ and $ρK$ two-body interactions respectively, as the clusters of the fixed center approximation. Then, we study the three-body $K -ρρ(f_2)$ and $ρ- ρK (K_1)$ interactions with the fixed center approximation of the Faddeev equations. Furthermore, we extrapolate the formalism to study the four-body, five-body and six-body systems, containing one $K$ meson and multi-$ρ$ mesons. In our research, without introducing any free parameters, we generate the $K_2 (1770)$ state in the three-body interaction with the mass of 1707 MeV and a width about 113 MeV, which are consistent with the experiments. We also find a clear resonant structure in our results of the five-body interaction, with a mass 2505 MeV and a width about 32 MeV or more, which is associated to the $K_4 (2500)$ state, where we obtain consistent results with the experimental findings. Furthermore, we predict some new states in the other many body interactions, $K_3 (2080)$, $K_5 (2670)$ (isospin $I=1/2$), and $K_4 (2640)$ (isospin $I=3/2$), with uncertainties.