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

TL;DR

This paper investigates genus two Heegaard splittings of knot exteriors, focusing on 1-genus 1-bridge knots and their unknotting tunnels, revealing conditions under which tunnels can be aligned with the boundary torus.

Contribution

It extends understanding of Heegaard splittings by analyzing cases where unknotting tunnels can be positioned on the boundary torus, beyond the classical 1-bridge knot scenario.

Findings

Unknotting tunnel can be levelled with the boundary torus.

Not all genus two splittings arise from 1-genus 1-bridge knots.

Conditions for positioning the tunnel on the boundary are identified.

Abstract

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.