Here is another way of distinguishing knots - the bridge number. A picture of a knot is a bit like a motorway; at each crossing there is an underpass and an overpass. One or more overpasses in succession form a bridge. A bridge starts from under one crossing, goes over other parts of the knot, and ends by diving under another crossing.
Our next pictures of knots show the number of bridges. Check that we have the numbers right!
5 bridges
6 bridges
7 bridges
There is no difficulty in making a picture of a knot with lots and lots of bridges, by twisting the knot to add many extra crossings. So for any given knot, we look for the picture of it which has the smallest number of bridges. That number of bridges is the bridge number of the knot. The bridge number of the unknot is 0. Also if a picture of a knot has only one bridge then the knot must be an unknot.
The last picture has two bridges and is a picture of the trefoil. The trefoil is knotted, so it has no picture with only one bridge. The bridge number of the trefoil is exactly 2.
What about the cinquefoil? Try redrawing it to see how few bridges
are needed. Can you be sure that someone else might not have drawn
it with fewer bridges?
Bridge numbers are simple to define, but hard to calculate. 
Invariants
© Mathematics and Knots, U.C.N.W.,Bangor, 1996 - 2002
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