Summary: UNIQUENESS OF BRIDGE SURFACES FOR 2-BRIDGE KNOTS
MARTIN SCHARLEMANN AND MAGGY TOMOVA
Abstract. We show that any 2-bridge knot in S3 has a bridge sphere from
which any other bridge surface can be obtained by stabilization, meridional
stabilization, perturbation and proper isotopy.
Establishing the uniqueness of Heegaard splittings for certain 3-manifolds has
been an interesting and surprisingly difficult problem. One of the earliest known
results was that of Waldhausen [Wa] who proved that S3
has a unique Heegaard
splitting up to stabilization. In [BoO], Bonahon and Otal proved that the same is
true of lens splaces (manifolds with a genus one Heegaard surface). A later proof
[RS] made use of the fact that any two weakly incompressible Heegaard splittings
of a manifold can be isotoped to intersect in a nonempty collection of curves that
are essential on both Heegaard surfaces.
There is an analogue to Heegaard splitting in the theory of links in 3-manifolds.
(By link, we include the possibility that K has one component, i. e. a knot is
a link.) Consider a link K in a closed orientable 3-manifold M with a Heegaard
surface P (i.e. M = A P B where A and B are handlebodies) and require that
each arc of K - P is P-parallel in M - P. We say that K is in bridge position