Heegaard splittings of knot exteriors

Abstract

The goal of this paper is to offer a comprehensive exposition of the current knowledge about Heegaard splittings of exteriors of knots in the 3-sphere. The exposition is done with a historical perspective as to how ideas developed and by whom. Several new notions are introduced and some facts about them are proved. In particular the concept of a 1/n-primitive meridian. It is then proved that if a knot K in S^3 has a 1/n-primitive meridian; then nK = K#...#K, n-times has a Heegaard splitting of genus nt(K) + n which has a 1-primitive meridian. That is, nK is mu-primitive.

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

References in corpus (4)

• Incompressible surfaces and (1,2)-knots
• On tunnel number one knots that are not (1,n)
• On the growth rate of tunnel number of knots
• Knots with g(E(K)) = 2 and g(E(K#K#K)) = 6 and Morimoto's Conjecture

Cited by in corpus (4)

• Stabilizations of Heegaard splittings of sufficiently complicated 3-manifolds (Preliminary Report)
• Heegaard splittings of 3--manifolds (Haifa 2005): Problems
• Heegaard splittings of sufficiently complicated 3-manifolds II: Amalgamation
• Knot exteriors with additive Heegaard genus and Morimoto's Conjecture