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A simple way to describe a knot projection. The advantage of this notation is that it enables a Knot Diagram to be drawn quickly.
For an oriented Alternating Knot with 
crossings, begin at an arbitrary crossing and label it 1. Now follow the
undergoing strand to the next crossing, and denote it 2. Continue around the knot following the same strand until each
crossing has been numbered twice. Each crossing will have one even number and one odd number, with the numbers running from
1 to 
.
Now write out the Odd Numbers 1, 3, ..., 
in a row, and underneath write the even crossing
number corresponding to each number. The Dowker Notation is this bottom row of numbers. When the sequence of even
numbers can be broken into two permutations of consecutive sequences (such as
), the knot is
composite and is not uniquely determined by the Dowker notation. Otherwise, the knot is prime and the Notation
uniquely defines a single knot (for amphichiral knots) or corresponds to a single knot or its Mirror Image (for chiral
knots).
For general nonalternating knots, the procedure is modified slightly by making the sign of the even numbers Positive if the 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.