Dowker Notation

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 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.

Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First  Knots." Math. Intell. 20, 33-48, Fall 1998.

Thistlethwaite, M. B. "Knot Tabulations and Related Topics." In Aspects of Topology in Memory of Hugh Dowker 1912-1982 (Ed. I. M. James and E. H. Kronheimer). Cambridge, England: Cambridge University Press, pp. 2-76, 1985.