 
Summary: UNIQUENESS OF BRIDGE SURFACES FOR 2BRIDGE KNOTS
MARTIN SCHARLEMANN AND MAGGY TOMOVA
Abstract. We show that any 2bridge knot in S3 has a bridge sphere from
which any other bridge surface can be obtained by stabilization, meridional
stabilization, perturbation and proper isotopy.
1. Introduction
Establishing the uniqueness of Heegaard splittings for certain 3manifolds 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 3manifolds.
(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 3manifold 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 Pparallel in M  P. We say that K is in bridge position
