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

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

2010-09-11

arxiv.org/abs/1009.2134v1

Mathematics

Geometric Topology

31 pages, 16 figures

Scientific paper

A knot K in a closed connected orientable 3-manifold M is called a 1-genus 1-bridge knot if (M,K) has a splitting into two pairs of a solid torus V_i (i=1,2) and a boundary parallel arc in it. The splitting induces a genus two Heegaard splitting of the exterior of K naturally, i.e., K has an unknotting tunnel. However the converse is not true in general. Then we study such general case in this paper. One of the conclusions is that the unknotting tunnel may be levelled with the torus \partial V_1=\partial V_2.

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