On nonsimple knots in lens spaces with tunnel number one

TL;DR

This paper proves that nonsimple prime tunnel number one knots in certain lens spaces are (1,1) knots, using advanced techniques in 3-manifold topology and Heegaard splittings.

Contribution

It establishes a classification result for nonsimple prime tunnel number one knots in lens spaces without Klein bottles, showing they are (1,1) knots.

Findings

Nonsimple prime tunnel number one knots in specified lens spaces are (1,1) knots.

Utilizes handle addition, Dehn filling, and Heegaard splitting techniques.

Provides structural insights into knots in 3-manifolds.

Abstract

A knot k in a closed orientable 3-manifold is called nonsimple if the exterior of k possesses a properly embedded essential surface of nonnegative Euler characteristic. We show that if k is a nonsimple prime tunnel number one knot in a lens space M (where M does not contain any embedded Klein bottles), then k is a (1,1) knot. Elements of the proof include handle addition and Dehn filling results/techniques of Jaco, Eudave-Munoz and Gordon as well as structure results of Schultens on the Heegaard splittings of graph manifolds.