Abby Thompson proved that if a link $K$ is in thin position but not in bridge
position then the knot complement contains an essential meridional planar
surface, and she asked whether some thin level surface must be essential. This
note is to give a positive answer to this question, showing that the if a link
is in thin position but not bridge position then a thinnest level surface is
essential. A theorem of Rieck and Sedgwick follows as a consequence, which says
that thin position of a connected sum of small knots comes in the obvious way