  In the present work we use three-body interaction formalism to investigate
the $K$-multi-$\rho$ interactions. First, we reproduced the resonances
$f_2(1270)$ and $K_1(1270)$ in the $\rho \rho$ and $\rho K$ two-body
interactions respectively, as the clusters of the fixed center approximation.
Then, we study the three-body $K -\rho \rho (f_2)$ and $\rho - \rho K (K_1)$
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-$\rho$ 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.
