Bridge Number

a 3 bridge presentation of the trefoilConsider the pictured trefoil knot K embedded in \mathbb{R}^3.  The xy plane is shown, and the intersections of the xy plane with K are marked in blue.  The knot is drawn with a dashed line where it lies below the xy plane; notice that the intersection of the lower half space with K consists of three unknotted arcs.  Similarly, the intersection of the upper half space with K is three unknotted arcs.  These arcs look a bit like bridges, and we say this is a three bridge presentation of K.

The smallest integer n such that a knot K has an n bridge presentation is called the bridge number of K.  The above picture shows that the bridge number of K 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.

A 2 bridge presentation of the trefoil

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