The simplest version of a knot can sometimes look very different from its everyday appearance. Take the bowline, a knot commonly used by sailors to make a loop in a rope end.
If we join the loose ends together, we can move the string around
through various positions until it crosses over itself only six
times. The six-crossing version of the bowline is the simplest
possible picture of this knot. We say that the bowline has a crossing number of 6.
It can be very difficult to tell apart two complicated knot diagrams.
Just four moves can make a knot unrecognisable. In 1899 a table
of knots was drawn up by C.M.Little. The two knots below were
listed as being different. It was only in 1974 that K.M.Perko
discovered that the two knots were one and the same. One task
facing mathematicains is to find fool proof ways of deciding whether
two knots are actually different or merely distorted versions
of one another.
Classification
© Mathematics and Knots, U.C.N.W.,Bangor, 1996 - 2002
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