  Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a
closed surface so that the complement of the surface is a connected essential
surface for the knot complement. In this paper, we examine some variations on
this question and prove it for all knots up to 11 crossings except for two
examples. We also establish the conjecture for all Montesinos knots and for all
generalized arborescently alternating knots. For knot exteriors containing
closed incompressible surfaces satisfying a simple homological condition, we
establish that the knots satisfy the Neuwirth conjecture. If there is a proper
degree one map from knot $K$ to knot $K'$ and $K'$ satisfies the Neuwirth
conjecture then we prove the same is true for knot $K$. Algorithms are given to
decide if a knot satisfies the various versions of the Neuwirth conjecture and
also the related conjectures about whether all non-trivial knots have essential
surfaces at integer boundary slopes.
