Knot Diagram

A knot diagram is a picture of a projection of a knot onto a plane. Usually, only double points are allowed (no more than two points are allowed to be superposed), and the double or crossing points must be "genuine crossings" which transverse in the plane. This means that double points must look like the above left diagram, and not the above right one. Also, it is usually demanded that a knot diagram contain the information if the crossings are overcrossings or undercrossings so that the original knot can be reconstructed.

The knot diagram of the trefoil knot is illustrated above.

Knot polynomials can be computed from knot diagrams. Such polynomials often (but not always) allow the knots corresponding to given diagrams to be uniquely identified.

Rolfsen (1976) gives a table of knot diagrams for knots up to 10 crossings and links up to four components and 9 crossings. Adams (1994) gives a smaller table of knots diagrams up to 9 crossings, two-component links up to 8 crossings, and three-component links up to 7 crossings. Livingston (1993) gives a list of diagrams for knots up to nine crossings.

REFERENCES:

Adams, C. C. "Table of Knots, Links, and Knot and Link Invariants." Appendix in The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 279-290, 1994.

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

Livingston, C. "Knot Table." Appendix 1 in Knot Theory. Washington, DC: Math. Assoc. Amer., pp. 221-228, 1993.

Rolfsen, D. "Table of Knots and Links." Appendix C in Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 388-429, 1976.