  We present a calculation of the contribution of the $\Theta$-term to the
neutron and proton electric dipole moments using seven 2+1+1-flavor HISQ
ensembles. We also estimate the topological susceptibility for the 2+1+1 theory
to be $\chi_Q = (66(9)(4) \rm MeV)^4$ in the continuum limit at $M_\pi = 135$
MeV. The calculation of the nucleon three-point function is done using
Wilson-clover valence quarks. The CP-violating form factor $F_3$ is calculated
by expanding in small $\Theta$. We show that lattice artifacts introduce a term
proportional to $a$ that does not vanish in the chiral limit, and we include
this in our chiral-continuum fits. A chiral perturbation theory analysis shows
that the $N(0) \pi(0)$ state should provide the leading excited state
contribution, and we study the effect of such a state. Detailed analysis of the
contributions to the neutron and proton electric dipole moment using two
strategies for removing excited state contamination are presented. Using the
excited state spectrum from fits to the two-point function, we find
$d_n^\Theta$ is small, $|d_n^\Theta| \lesssim 0.01 \overline \Theta e$ fm,
whereas for the proton we get $|d_p^\Theta| \sim 0.02 \overline \Theta e$ fm.
On the other hand, if the dominant excited-state contribution is from the $N
\pi$ state, then $|d_n^\Theta|$ could be as large as $0.05 \overline \Theta e$
fm and $|d_p^\Theta| \sim 0.07 \overline \Theta e$ fm. Our overall conclusion
is that present lattice QCD calculations do not provide a reliable estimate of
the contribution of the $\Theta$-term to the nucleon electric dipole moments,
and a factor of ten higher statistics data are needed to get better control
over the systematics and possibly a $3\sigma$ result.
