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A regular circle-valued Morse function on the knot complement CK = S3 ∖ K is a
function f : CK → S1 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) = S3 ∖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 nonisotopic incompressible Seifert
surface. We also analyze the behavior of the circular width under some knot
operations.
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