• DOI:10.1007/BF03025227
• Corpus ID: 18027155

The first 1,701,936 knots

@article{Hoste1998TheF1,
  title={The first 1,701,936 knots},
  author={Jimmy-John O. E. Hoste and Morwen Thistlethwaite and Jeffrey R. Weeks},
  journal={The Mathematical Intelligencer},
  year={1998},
  volume={20},
  pages={33-48}
}

• J. Hoste, M. Thistlethwaite, J. Weeks
• Published 1 September 1998
• Computer Science
• The Mathematical Intelligencer

inc lude all pr ime knots wi th 16 or fewer crossings. This r epresen t s more than a 130-fold increase in the number of t abu la ted knots s ince the last burs t of tabula t ion tha t t ook p lace in the early 1980s. With more than 1.7 mil l ion knots now in the tables, we hope that the census will serve as a r ich source of examples and coun te rexamples and as a genera l test ing ground for our collective intuition. To this end, we have wri t ten a UNIX-based compute r p rog ram cal led KnotScape which a l lows easy access to the tables. The account of our me thodo logy is p re faced by a br ie f h is tory of knot tabulat ion, concentra t ing most ly on events taking p lace within the last 30 years. The survey art icle [Thil] contains fur ther detai ls on the work of the n ine teenth-century tabula tors , but, above all, the r eade r is encouraged to consul t the original sources, in par t icu la r the excel lent ser ies of pape r s by Tait [Tail. Ki rkman 's papers make fascinat ing reading, as they abound with original ideas and ornate l anguage -h i s definit ion of the te rm "knot' is a single sentence of 101 words. Conway's landmark p a p e r [Con] is also highly r ecommended . An impor tant feature of our pro jec t is that we have worked in two complete ly separa te teams, producing two tabula t ions which were kept secre t unti l af ter they were complete. 1 Al though it would be foolhardy to claim with absolute cer ta in ty that our tables a re correct , we must repor t the gratifying exper ience of fmding that our lists of 1,701,936 knots were in comple te agreement! Moreover, we did not use exac t ly the same methods , the pr imary difference being the use of hyperbol ic geomet ry by Hoste and Weeks and the comple te absence of hyperbol ic invariants in Thist lethwaite 's approach. Nevertheless, our overall programs are similar in spirit and differ little from the methods of most tabula tors who precede us. As par t of our tabulation, using Weeks 's program SnapPea we were able to compute the symmetry groups of the knots; we have included a short in t roduct ion to this beautiful and intriguing topic.