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Suppose a link K in a 3-manifold M is in bridge position with respect to twodifferent bridge surfaces P and Q, both of which are c-weakly incompressible inthe complement of K. Then either  P and Q can be properly isotoped to intersect in a nonempty collection ofcurves that are essential (including non-meridional) on both surfaces, or  K is a Conway product with respect to an incompressible Conway sphere thatnaturally decomposes both P and Q into bridge surfaces for the respectivefactor link(s).

Conway products and links with multiple bridge surfaces