Composite Knot

A composite knot is a knot that is not a prime knot. Schubert (1949) showed that every knot can be uniquely decomposed (up to the order in which the decomposition is performed) as a knot sum of a class of knots known as prime knots, which cannot themselves be further decomposed (Hoste et al. 1998).

Knots that make up a knot sum of a composite knot are known as factor knots.

Combining prime knots gives no new knots for knots of three to five crossing, but two additional composite knots (the granny knot and square knot) with six crossings. The granny knot is the knot sum of two trefoils with the same chirality (), while the square knot is the knot sum of two trefoils with opposite chiralities (). There is a single composite knot of seven crossings () and four composite knots of eight crossings (, , , and ). The numbers of composite knots having , 2, ... crossings are therefore 0, 0, 0, 0, 0, 2, 1, 4, ....

REFERENCES:

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 8, 1994.

Cipra, B. A. "To Have and Have Knot: When Are Knots Alike?" Science 241, 1291-1292, 1988.

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

Schubert, H. Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturwiss. Klasse, 3rd Abhandlung. 1949.