 
Summary: CONWAY PRODUCTS AND LINKS WITH MULTIPLE BRIDGE
SURFACES
MARTIN SCHARLEMANN AND MAGGY TOMOVA
Abstract. Suppose a link K in a 3manifold M is in bridge position with
respect to two different bridge surfaces P and Q, both of which are cweakly
incompressible in the complement of K. Then either
· P and Q can be properly isotoped to intersect in a nonempty collection
of curves that are essential on both surfaces, or
· K is a Conway product with respect to an incompressible Conway sphere
that naturally decomposes both P and Q into bridge surfaces for the
respective factor link(s).
1. Introduction
A link K in a 3manifold M is said to be in bridge position with respect to a
Heegaard surface P for M if each arc of K  P is parallel to P. P is then called
a bridge surface for K in M. Given a link in bridge position with respect to P, it
is easy to construct more complex bridge surfaces for K from P; for example by
stabilizing the Heegaard surface P or by perturbing K to introduce a minimum
and an adjacent maximum. As with Heegaard splitting surfaces for a manifold, it
is likely that most links have multiple bridge surfaces even apart from these simple
operations. In an effort to understand how two bridge surfaces for the same link
