  We give a simple sufficient condition for a spun-normal surface in an ideal
triangulation to be incompressible, namely that it is a vertex surface with
non-empty boundary which has a quadrilateral in each tetrahedron. While this
condition is far from being necessary, it is powerful enough to give two new
results: the existence of alternating knots with non-integer boundary slopes,
and a proof of the Slope Conjecture for a large class of 2-fusion knots. While
the condition and conclusion are purely topological, the proof uses the
Culler-Shalen theory of essential surfaces arising from ideal points of the
character variety, as reinterpreted by Thurston and Yoshida. The criterion
itself comes from the work of Kabaya, which we place into the language of
normal surface theory. This allows the criterion to be easily applied, and
gives the framework for proving that the surface is incompressible. We also
explore which spun-normal surfaces arise from ideal points of the deformation
variety. In particular, we give an example where no vertex or fundamental
surface arises in this way.
