  A regular circle-valued Morse function on the knot complement C(K) = S^3\K is
a function f from C(K) to S^1 which separates critical points and which behaves
nicely in a neighborhood of the knot. Such a function induces a handle
decomposition on the knot exterior E(K) = S^3\N (K), with the property that
every regular level surface contains a Seifert surface for the knot. We
rearrange the handles in such a way that the regular surfaces are as simple as
possible. To make this precise the concept of circular width for E(K) is
introduced. When E(K) is endowed with a handle decomposition which realizes the
circular width we will say that the knot K is in circular thin position. We use
this to show that many knots have more than one non-isotopic incompressible
Seifert surface. We also analyze the behavior of the circular width under some
knot operations.
