Prediction of possible $DK_1$ bound states

Stimulated by recent experimental observation of $X_1(2900)$ just below the $\bar D_1K$ threshold, we extend our previous study of $\bar D_1D$ S-wave bound state by vector meson exchange to $\bar D_1K$ system as well as similar $\bar DK_1$, $D_1K$ and $DK_1$ systems to look for possible bound states. We find that the potential of $D K_1$ is attractive and strong enough to form bound states with mass around 3110 MeV for $DK_1(1270)$ and 3240 MeV for $DK_1(1400)$. $D_1 K$ is also attractive but weaker, hardly enough to form bound states. While $\bar DK_1$ becomes further less attractive, the potential between $\bar D_1 K$ is the weakest, definitely too weak to form any bound state, which excludes the recently observed $X_1(2900)$ to be a $\bar D_1 K$ bound state. We also give the decay properties of the predicted $D K_1$ bound states.