Circular thin position for knots in the 3-sphere

Abstract

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.

25 pages, 10 figures

Cited by in corpus (6)

• Thin position for knots, links, and graphs in 3-manifolds
• Additivity of Handle Number and Morse-Novikov Number of a-small Knots
• Additivity of Circular Width
• The Morse-Novikov number of knots under connected sum and cabling
• A Distance for Circular Heegaard Splittings
• Circular handle decompositions of free genus one knots