Tabulating an enormous assortment of knots.
Two new catalogs, produced independently, tabulate all knotted loops of a one-dimensional string having 16 or fewer crossings when the knot is laid flat. That's 1,701,936 different knots.
Jim Hoste of Pitzer College in Claremont, Calif., and Jeffrey R. Weeks of Canton, N.Y., worked on one listing, while Morwen Thistlethwaite of the University of Tennessee in Knoxville generated the other. They jointly describe their tabulation efforts in the fall issue of Mathematical Intelligencer.
"With more than 1.7 million knots now in the tables, we hope that the census will serve as a rich source of examples and counterexamples and as a general testing ground for our collective intuition," the researchers say.
Mathematicians typically concern them selves with knots having their two ends connected to form a loop. One way to characterize such a knot is to lay it flat and determine its crossing number by counting the minimum number of times one part of the loop crosses over or under another part.
Efforts to tabulate knots began about 120 years ago, after British physicist Lord Kelvin hypothesized that atoms could be described as vortices in the ether, an intangible fluid then thought to fill all space. He proposed that different elements would correspond to vortices bent into different types of knotted tubes forming closed loops.
Inspired by this theory, fellow physicist Peter G. Tait began investigating knots and produced the first knot tables, organized according to crossing number.
Tait enumerated all possible knot diagrams up to a given crossing number, then grouped those diagrams representing the same knot type. He stopped at knots with seven crossings--a total of 15 knot types.
Using a similar strategy and with the help of computers, Hoste, Weeks, and Thistlethwaite produced independent tabulations of knots with 16 or fewer crossings. Kept secret until they were finished, the two lists were in complete agreement.
Hoste and Thistlethwaite are now preparing the 17-crossing list.
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|Title Annotation:||mathematicians calculate 1.7 million possible knots|
|Article Type:||Brief Article|
|Date:||Oct 10, 1998|
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