Dark energy with polytropic equation of state

Equation of state parameter plays a significant role for guessing the real nature of dark energy. In the present paper polytropic equation of state $p=\omega\rho^n$ is chosen for some of the kinematical $\Lambda$-models viz., $\Lambda \sim (\dot a/a)^2$, $\Lambda \sim \ddot a/a$ and $\Lambda \sim \rho$. Although in dust cases ($\omega=0$) closed form solutions show no dependency on the polytropic index $n$, but in non-dust situations some new possibilities are opened up including phantom energy with supernegative ($\omega<-1$) equation of state parameter.