Consider the pictured trefoil knot embedded in . The plane is shown, and the intersections of the plane with are marked in blue. The knot is drawn with a dashed line where it lies below the plane; notice that the intersection of the lower half space with consists of three unknotted arcs. Similarly, the intersection of the upper half space with is three unknotted arcs. These arcs look a bit like bridges, and we say this is a three bridge presentation of .

The smallest integer such that a knot has an bridge presentation is called the **bridge number** of . The above picture shows that the bridge number of is less than or equal to three. We can do better though! Below is a two bridge presentation of the trefoil. Since a bridge number one knot is the unknot, this shows that the trefoil has bridge number two.

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