Dowker Notation

A simple way to describe a knot projection. The advantage of this notation is that it enables a Knot Diagram tobe drawn quickly.

For an oriented Alternating Knot with crossings, begin at an arbitrary crossing and label it 1. Now follow theundergoing strand to the next crossing, and denote it 2. Continue around the knot following the same strand until eachcrossing has been numbered twice. Each crossing will have one even number and one odd number, with the numbers running from1 to .

Now write out the Odd Numbers 1, 3, ..., in a row, and underneath write the even crossingnumber corresponding to each number. The Dowker Notation is this bottom row of numbers. When the sequence of evennumbers can be broken into two permutations of consecutive sequences (such as ), the knot iscomposite and is not uniquely determined by the Dowker notation. Otherwise, the knot is prime and the Notationuniquely defines a single knot (for amphichiral knots) or corresponds to a single knot or its Mirror Image (for chiralknots).

For general nonalternating knots, the procedure is modified slightly by making the sign of the even numbers Positive ifthe crossing is on the top strand, and Negative if it is on the bottom strand.

These data are available only for knots, but not for links, from Berkeley's gopher site.

References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 35-40, 1994.

Dowker, C. H. and Thistlethwaite, M. B. ``Classification of Knot Projections.'' Topol. Appl. 16, 19-31, 1983.

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