Low Dimensional Topology

July 9, 2008

Bridge positions and branched covers

Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 10:35 am

Recall that every 2-sphere in the 3-sphere cuts it into two balls.  A knot K in the 3-sphere is in bridge position with respect to this sphere if it intersects each ball in a collection of boundary parallel arcs.  One of the nice things about bridge position is that in the double branched cover over the knot, the bridge sphere lifts to a Heegaard surface whose genus is one less than the number of arcs (i.e. bridges) in each ball.  Thus the Heegaard genus of the double branched cover is at most the bridge number of the knot minus one.

Of course, not every Heegaard splitting for the double branched cover comes from such a construction, so it’s possible that the Heegaard genus is much lower than the bound given by the bridge number.  For example, Scott Taylor has pointed out that a Heegaard surface for the knot complement also lifts to a Heegaard surface for the double branched cover.  The genus of the lifted surface is one less than twice the original genus.  Thus if the Heegaard genus of the knot complement is much smaller than the bridge number, the lift of the bridge surface will not be minimal genus.  This is the case, for example, with torus knots; they all have Heegaard genus two, but their bridge numbers can be arbitrarily large.  (One can apply a similar argument to higher genus bridge surfaces, in which case the examples of Moriah, Minsky and Schleimer [1] or my examples with Thompson [2] for genus one, can be used.)

Of course, just because these lifted surfaces are not minimal genus does not mean they’re stabilized/reducible.  For example, the double branched cover of a torus knot is a Seifert fibered space.  Some Seifert fibered spaces have strongly irreducible, non-minimal genus Heegaard splittings called horizontal splittings.  It’s possible that the bridge surface lifts to an irreducible horizontal surface.  (The way the bridge surface intersects the fibering of the knot complement makes this plausible, though I haven’t checked this carefully.)  Thus one can still ask when the non-minimal Heegaard surfaces that come from lifting a minimal bridge surface to the double branched cover are in fact reducible.

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