  The parameter $\epsilon_K$ measures $CP$ violation in the neutral kaon
system. It is a sensitive probe of new physics and plays a prominent role in
the global fit of the Cabibbo-Kobabyashi-Maskawa matrix. The perturbative
theory uncertainty is currently dominated by the top-quark contribution. Here,
we present the calculation of the full two-loop electroweak corrections to the
top-quark contribution to $\epsilon_K$, including the resummation of QED-QCD
logarithms. We discuss different renormalization prescriptions for the
electroweak input parameters. In the traditional normalization of the weak
Hamiltonian with two powers of the Fermi constant $G_F$, the top-quark
contribution is shifted by $-1\%$.
