  We report the first high spatial resolution measurement of magnetic fields
surrounding LkH$\alpha$ 101, a part of the Auriga-California molecular cloud.
The observations were taken with the POL-2 polarimeter on the James Clerk
Maxwell Telescope within the framework of the B-fields In Star-forming Region
Observations (BISTRO) survey. Observed polarization of thermal dust emission at
850 $\mu$m is found to be mostly associated with the red-shifted gas component
of the cloud. The magnetic field displays a relatively complex morphology. Two
variants of the Davis-Chandrasekhar-Fermi method, unsharp masking and structure
function, are used to calculate the strength of magnetic fields in the plane of
the sky, yielding a similar result of $B_{\rm POS}\sim 115$ $\mathrm{\mu}$G.
The mass-to-magnetic-flux ratio in critical value units, $\lambda\sim0.3$, is
the smallest among the values obtained for other regions surveyed by POL-2.
This implies that the LkH$\alpha$ 101 region is sub-critical and the magnetic
field is strong enough to prevent gravitational collapse. The inferred $\delta
B/B_0\sim 0.3$ implies that the large scale component of the magnetic field
dominates the turbulent one. The variation of the polarization fraction with
total emission intensity can be fitted by a power-law with an index of
$\alpha=0.82\pm0.03$, which lies in the range previously reported for molecular
clouds. We find that the polarization fraction decreases rapidly with proximity
to the only early B star (LkH$\alpha$ 101) in the region. The magnetic field
tangling and the joint effect of grain alignment and rotational disruption by
radiative torques are potential of explaining such a decreasing trend.
