Circular thin position for knots in $S^3$

Abstract

A regular circle-valued Morse function on the knot complement $C_K=S^3\setminus K$ is a function $f:C_K\to S^1$ 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 $E\left(K\right)=S^3\setminus N\left(K\right)$, 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 $E\left(K\right)$ is introduced. When $E\left(K\right)$ 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.