# Mathematics Colloquia and Seminars

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### 3-manifolds with planar presentations and the width of satellite knots

**Geometry/Topology**

Speaker: | Martin Scharlemann, UC Santa Barbara |

Location: | 693 Kerr |

Start time: | Tue, May 6 2003, 3:10PM |

We consider compact 3-manifolds M having a submersion h to R in which each generic point inverse is a planar surface. The standard height function on a submanifold of the 3-sphere is a motivating example. To (M, h) we associate a connectivity graph G. For M in the 3-sphere, G is a tree if and only if there is a Fox reimbedding of M which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of the complement of M is a tree, then there is a level-preserving reimbedding of M so that its complement is a connected sum of handlebodies. Corollary: The width of a satellite knot is no less than the width of its pattern knot. In particular, the width of the sum K # L of knots is no less than the maximum of the widths of K and L.

This is part of the BATS conference.

Prof. Scharlemann will also give three talks on Heegaards surfaces. Monday at noon, Tuesday at 11AM and Wed. at 4PM.

There is a dinner following the BATS conference.