  We consider the Kisin variety associated to a $n$-dimensional absolutely
irreducible mod $p$ Galois representation $\bar\rho$ of a $p$-adic field $K$
and a cocharacter $\mu$. Kisin conjectured that the Kisin variety is connected
in this case. We show that Kisin's conjecture holds if $K$ is totally ramfied
with $n=3$ or $\mu$ is of a very particular form. As an application, we also
get a connectedness result for the deformation ring associated to $\bar\rho$ of
given Hodge-Tate weights. We also give counterexamples to show Kisin's
conjecture does not hold in general.
