### Bridge Number

Consider 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.