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Virtual knot theory is an extension of knot theory which allows for virtual crossings, which are not over or under, in a diagram. In addition to the Reidemeister moves for classical crossings, moves involving virtual crossings are allowed. For details, see Louis Kauffman, Virtual Knot Theory, Europ. J. Combinatorics (1999) 20, 663-691, arXiv:math.GT/9811028.

This table of virtual knots was generated using a computer program written by Jeremy Green under the supervision of Dror Bar-Natan. The program generates a list of Gauss codes, then, via Reidemeister moves, determines which are equivalent to each other. Using an Athlon XP 2500+ with 512 MB RAM, the enumeration can be done up to 8 crossing oriented virtual knots, using 10 GB of disk, in about 24 hours. This 8 crossing enumeration is only useful for knots up to 6 crossings because of the need for higher crossing intermediates when finding relationships. This enumeration counted 725854 oriented virtual knots with up to 6 crossings, or 92800 when counting inverses and mirror images as the same knot.

The program also generates all of the content on the individual pages for the virtual knots, including the drawings and the various invariants. It is available here, in C source code form. Instructions are in the included README file. It was run on Red Hat Linux 9, but is fairly portable. For drawing to a PNG image, it requires Ghostscript and Imagemagick to be installed.

See also: Knotilus has lists and drawings of virtual knots as the result of collaboration of Ralph Furmaniak and Louis Kauffman