  The influence of hyperfine interactions on the tune-out wavelengths of the
$^{39,40,41}$K isotopes of the potassium atom was investigated. The hyperfine
interaction of the $4s_{1/2}$ state result in a shift and splitting of the
primary tune-out wavelength near 769 nm. The $4s_{1/2}$ ground state hyperfine
splittings of the primary tune-out were almost equal to the hyperfine
splittings of the ground states. The splittings in the wavelengths were 0.0008,
0.0027 and 0.0005 nm for $^{39}$K, $^{40}$K and $^{41}$K respectively. The
hyperfine splitting of the $np_{J}$ levels leads to the creation of additional
tune-out wavelengths. The additional tune-out wavelengths could be difficult to
detect due to very small differences from the transition wavelengths to the
$4p_{J,F}$ states. The hyperfine Stark shift for the ground states of all three
isotopes were also computed and the value for $^{30}$K was found to be
compatible with the previous experiments and the most recent calculation using
relativistic many body perturbation theory.
