Circular thin position for knots in the 3-sphere

TL;DR

This paper introduces the concept of circular thin position for knots in the 3-sphere, providing a new way to analyze Seifert surfaces and handle decompositions, revealing multiple incompressible Seifert surfaces for many knots.

Contribution

It defines circular width and circular thin position for knots, and demonstrates their applications in understanding Seifert surfaces and knot operations.

Findings

Many knots have multiple non-isotopic incompressible Seifert surfaces.

Circular width can be used to analyze the complexity of knot complements.

Handle rearrangements can simplify the structure of knot exteriors.

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.