Genus two Heegaard splittings of exteriors of 1-genus 1-bridge knots II

Details Genus two Heegaard splittings of exteriors of 1-genus 1-bridge knots II Genus two Heegaard splittings of exteriors of 1-genus 1-bridge knots II

2010-09-13

arxiv.org/abs/1009.2279v1

Mathematics

Geometric Topology

21 pages, 7 figures

Scientific paper

A knot K is called a 1-genus 1-bridge knot in a 3-manifold M if (M,K) has a Heegaard splitting (V_1,t_1)\cup (V_2,t_2) where V_i is a solid torus and t_i is a boundary parallel arc properly embedded in V_i. If the exterior of a knot has a genus 2 Heegaard splitting, we say that the knot has an unknotting tunnel. Naturally the exterior of a 1-genus 1-bridge knot K allows a genus 2 Heegaard splitting, i.e., K has an unknotting tunnel. But, in general, there are unknotting tunnels which are not derived form this procedure. Some of them may be levelled with the torus \partial V_1=\partial V_2, whose case was studied in our previous paper. In this paper, we consider the remaining case.

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