Knot Theory - Tabulating Knots

Tabulating Knots

Traditionally, knots have been catalogued in terms of crossing number. Knot tables generally include only prime knots and only one entry for a knot and its mirror image (even if they are different) (Hoste, Thistlethwaite & Weeks 1998). The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult (Hoste 2005, p. 20). Tabulation efforts have succeeded in enumerating over 6 billion knots and links (Hoste 2005, p. 28). The sequence of the number of prime knots of a given crossing number, up to crossing number 16, is 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705... (sequence A002863 in OEIS). While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing (Adams 2004).

The first knot tables by Tait, Little, and Kirkman used knot diagrams, although Tait also used a precursor to the Dowker notation. Different notations have been invented for knots which allow more efficient tabulation (Hoste 2005).

The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings (Hoste, Thistlethwaite & Weeks 1998). The development of knot theory due to Alexander, Reidemeister, Seifert, and others eased the task of verification and tables of knots up to and including 9 crossings were published by Alexander–Briggs and Reidemeister in the late 1920s.

The first major verification of this work was done in the 1960s by John Horton Conway, who not only developed a new notation but also the Alexander–Conway polynomial (Conway 1970)(Doll & Hoste 1991). This verified the list of knots of at most 11 crossings and a new list of links up to 10 crossings. Conway found a number of omissions but only one duplication in the Tait–Little tables; however he missed the duplicates called the Perko pair, which would only be noticed in 1974 by Kenneth Perko (Perko 1974). This famous error would propagate when Dale Rolfsen added a knot table in his influential text, based on Conway's work.

In the late 1990s Hoste, Thistlethwaite, and Weeks tabulated all the knots through 16 crossings (Hoste, Thistlethwaite & Weeks 1998). In 2003 Rankin, Flint, and Schermann, tabulated the alternating knots through 22 crossings (Hoste 2005).