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### Bridge Number and the Curve Complex

**Geometry/Topology**

Speaker: | Jesse Johnson, UC Davis |

Location: | 2112 MSB |

Start time: | Wed, Apr 19 2006, 4:10PM |

An unknotting tunnel for a knot in the three-sphere determines a simple closed curve in the standard genus-two Heegaard splitting of S^3. This suggests a definition, via the curve complex, for the "distance" of the unknotting tunnel. I will show how this distance is related to the Seifert genus of the knot, the existence of alternate tunnels, and in particular the bridge number of the knot. I will also describe how the geometry of the curve complex can be used to prove the existence of knots with arbitrarily high distance tunnels.