  We report on a study of modulation transfer spectroscopy of the
$4\textrm{S}_{1/2}\rightarrow 4\textrm{P}_{1/2}$ ($D_{1}$) transition of
naturally abundant potassium in a room-temperature vapour cell. This transition
is critical for laser cooling and optical pumping of potassium and our study is
therefore motivated by the need for robust laser frequency stabilisation.
Despite the absence of a closed transition, the small ground-state hyperfine
splitting in potassium results in strong crossover features in the $D_{1}$
modulation transfer spectrum. To emphasise this we compare the $D_{1}$ and
$D_{2}$ spectra of potassium with those of rubidium. Further, we compare our
experimental results with a detailed theoretical simulation, examining
different pump-probe polarization configurations to identify the optimal
signals for laser frequency stabilisation. We find good agreement between the
experiment and the theory, especially for the $\textrm{lin} \parallel
\textrm{lin}$ polarization configuration.
