2,015 research outputs found
Surface bundles with genus two Heegaard splittings
It is known that there are surface bundles of arbitrarily high genus which
have genus two Heegaard splittings. The simplest examples are Seifert fibered
spaces with the sphere as a base space, three exceptional fibers and which
allow horizontal surfaces. We characterize the monodromy maps of all surface
bundles with genus two Heegaard splittings and show that each is the result of
integral Dehn surgery in one of these Seifert fibered spaces along loops where
the Heegaard surface intersects a horizontal surface. (This type of surgery
preserves both the bundle structure and the Heegaard splitting.)Comment: 30 pages, 8 figure
Bridge Number and the Curve Complex
We show that there are hyperbolic tunnel-number one knots with arbitrarily
high bridge number and that "most" tunnel-number one knots are not one-bridge
with respect to an unknotted torus. The proof relies on a connection between
bridge number and a certain distance in the curve complex of a genus-two
surface.Comment: 13 pages, 3 figures. References have been added and the order of
exposition changed slightl
Locally unknotted spines of Heegaard splittings
We show that under reasonable conditions, the spines of the handlebodies of a
strongly irreducible Heegaard splitting will intersect a closed ball in a graph
which is isotopic into the boundary of the ball. This is in some sense a
generalization of the results by Scharlemann on how a strongly irreducible
Heegaard splitting surface can intersect a ball.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-63.abs.htm
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