  We present state-of-the-art results for the matrix elements of flavor
diagonal tensor operators within the nucleon state. The calculation of the
dominant connected contribution is done using eleven ensembles of gauge
configurations generated by the MILC Collaboration using the highly improved
staggered quark (HISQ) action with 2+1+1 dynamical flavors. The calculation of
the disconnected contributions is done using seven (six) ensembles for the
strange (light) quarks. These high-statistics simulations allowed us to address
various systematic uncertainties. A simultaneous fit in the lattice spacing and
the light-quark mass is used to extract the tensor charges in the continuum
limit and at $M_\pi=135$ MeV. Results for the proton in the $\overline{MS}$
scheme at 2~GeV are: $g_T^u = 0.784(28)(10)$, $g_T^d = -0.204(11)(10)$ and
$g_T^s = -0.0027(16)$. Implications of these results for constraining the quark
electric dipole moments and their contributions to the neutron electric dipole
moment are discussed.
