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Uniqueness of bridge surfaces for 2-bridge knots. (English) Zbl 1152.57006
Considering a link $$K$$ in a closed 3-manifold $$M$$ leads to a generalization of bridge presentations for $$K\subset S^3$$. Instead of the 2-sphere instrumental in such a presentation one uses a Heegard surface for $$M$$ (bridge surface). The question of uniqueness of a Heegard splitting (up to stabilization) then corresponds to the uniqueness of bridge surfaces up to certain trivial changes of the bridge surface and the link. In addition of the stabilization of the Heegard splitting there is a “perturbation move” and a “meridional stabilization” taking account of the link $$K\subset S^3$$. The main result is: Every 2-bridge knot in $$S^3$$ has a unique bridge sphere, in the sense that any other bridge surface can be obtained by the changes mentioned above and proper isotopy.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
bridge surface; Heegaard splitting; 2-bridge knots
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##### References:
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