### University of Arkansas Topology Seminar: 2/20/2014

### Speaker: Sean Bowman

### Title: Bridge numbers of knots in the page of an open book

Abstract: An open book decomposition of a
3−manifold *M* consists of a binding *L* and a bundle
map from the complement of *L* to the circle such that the
preimage of a point of the circle is the interior of a surface
*S* whose boundary is the link *L*. Such a surface is called
a page of the open book, and we examine knots lying in these
surfaces. By studying the action of the map on the arc and curve
complex of a page induced by the bundle map, we find knots which have
arbitrarily large bridge number with respect to Heegaard splittings of
every genus up to minus the Euler characteristic of *S*. Several
corollaries will be noted.

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