Incompressible surfaces and (1,2)-knots

Abstract

We give a description of all (1,2)-knots in S^3 which admit a closed meridionally incompressible surface of genus 2 in their complement. That is, we give several constructions of (1,2)-knots having a meridionally incompressible surface of genus 2, and then show that any such surface for a (1,2)-knot must come from one of the constructions. As an application, we show explicit examples of tunnel number one knots which are not (1,2)-knots.

This is the version published by Geometry & Topology Monographs on 3 December 2007

References in corpus (1)

• High distance knots

Cited by in corpus (6)

• High distance knots
• Heegaard splittings of knot exteriors
• Tunnel number degeneration under the connected sum of prime knots
• Heegaard splittings of 3--manifolds (Haifa 2005): Problems
• On nonsimple knots in lens spaces with tunnel number one
• Hyperbolic (1,2)-knots in S^3 with crosscap number two and tunnel number one