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} }
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…
277 Citations
The 250 Knots with up to 10 Crossings
- Computer Science
- 2017
A way to generate the Rolfsen table in a simple, clear, and reproducible manner by generating all planar knot diagrams with up to 10 crossings and applying several simplifications to group the knot diagrams into equivalence classes.
Examples related to the crossing number, writhe, and maximal bridge length of knot diagrams
- Mathematics
- 2003
The “Perko pair” knot 10161 = 10162 [R, p. 415; identification noted in second printing] is exceptional in at least two distinct ways. It first achieved its name and fame when Perko [P] discovered…
Conway’s Knotty Past
- MathematicsThe Mathematical Intelligencer
- 2021
J ohn Conway’s impact has been felt far across mathematics. His interests were wide, and through his often playful approach, he uncovered fundamental theories that allowed for whole areas of research…
Tait’s conjectures and odd crossing number amphicheiral knots
- Mathematics
- 2007
We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his pioneering work in tabulating the simplest knots, and solved a century later using the Jones…
Introduction 1. Motivating Ideas
- Mathematics
At its core, this thesis is a study of knots, objects which human beings encounter with extraordinary frequency. We may find them on our person (in our shoelaces and neckties), or around our homes…
Sampling Lissajous and Fourier Knots
- MathematicsExp. Math.
- 2009
Several theorems are proved that allow us to place bounds on the number of Lissajous knot types with given frequencies and to efficiently sample all possible Lissjous knots with a given set of frequencies.
The modern study of knots grew out an attempt by three 19th-century Scottish physicists to apply knot theory to fundamental questions about the universe
- Physics
- 2016
T ake a length of rope, loop and weave it around itself and connect its ends. The result , of course, is a knot. Creating a knot seems simple, yet knot theory is one of the most active fields in…
Enumerating the Prime Alternating Knots, Part I
- Mathematics
- 2002
1. Abstract The enumeration of prime knots has a long and storied history, beginning with the work of T. P. Kirkman [9,10], C. N. Little [14], and P. G. Tait [19] in the late 1800’s, and continuing…
Knots in knots: A study of classical knot diagrams
- Mathematics
- 2016
The structure of classical minimal prime knot presentations suggests that there are often, perhaps always, subsegments that present either the trefoil or the figure-eight knot. A comprehensive study…
Generating Infinite Links as Periodic Tilings of the da Vinci–Dürer Knots
- ArtThe Mathematical Intelligencer
- 2017
There are six woodcuts of Albert Dürer’s black discs, upon which a symmetrical and concentrically arranged arabesque scrollwork of ribbons or festoons stands out in relief. These wonderful…
References
SHOWING 1-10 OF 72 REFERENCES
IX.—Alternate ± Knots of Order Eleven
- MathematicsTransactions of the Royal Society of Edinburgh
- 1892
1. A year ago last April, Prof. Tait proposed that I should undertake to derive from Mr Kirkman's polyhedral drawings the alternate ± knots of eleven crossings, thus doing for order 11 what had been…
On the classification of knots
- Mathematics
- 1974
Linking numbers between branch curves of irregular covering spaces of knots are used to extend the classification of knots through ten crossings and to show that the only amphicheirals in…
XXVI.—The 364 Unifilar Knots of Ten Crossings, Enumerated and Described
- HistoryTransactions of the Royal Society of Edinburgh
- 1886
The 119 subsolids (marked ss) and the 244 unsolids (marked us), of these unifilars are here arranged in lists according to their flaps. Fe is the number of flaps of e loops upon a knot; and the…
XXX.—Non-Alternate ± Knots
- MathematicsTransactions of the Royal Society of Edinburgh
- 1900
1. The following paper is a contribution to the theory of non-alternate ± knots, together with a census of these knots for Order Ten; that is, all the knots are given which have in reduced form just…
A tabulation of oriented links
- Computer Science, Mathematics
- 1991
This paper enumerates all prime, nonsplit, oriented, classical links having two or more components and nine or fewer crossings and relies heavily on the HOMFLY and Kauffman polynomials to distinguish inequivalent links.
XI.—On Knots, with a Census of the Amphicheirals with Twelve Crossings
- MathematicsTransactions of the Royal Society of Edinburgh
- 1918
The theory of the knotting of curves, except for a few elementary theorems due to Listing, was entirely neglected until Tait was led to a consideration of knots by Sir W. Thomson's (Lord Kelvin's)…
The rate of growth of the number of prime alternating links and tangles
- Mathematics
- 1998
When introduced to the subject of knot theory, it is natural to ask how the number of knots and links grows in relation to crossing number. The purpose of this article is to address this question for…
XVII.—The Enumeration, Description, and Construction of Knots of Fewer than Ten Crossings
- ArtTransactions of the Royal Society of Edinburgh
- 1884
1. By a knot of n crossings, I understand a reticulation of any number of meshes of two or more edges, whose summits, all tessaraces (ἀκή), are each a single crossing, as when you cross your…
XVIII.—Non-Alternate ± Knots, of Orders Eight and Nine
- MathematicsTransactions of the Royal Society of Edinburgh
- 1890
1. To complete the census of knots of any given order, that is, minimum number of crossings, it is necessary to include not only those in which the crossings are taken alternately over and under…