Mathematics – Geometric Topology
Scientific paper
2008-10-21
Mathematics
Geometric Topology
25 pages, 10 figures
Scientific paper
10.2140/agt.2009.9.429
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.
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