The first 1,701,936 knots

  title={The first 1,701,936 knots},
  author={Jimmy-John O. E. Hoste and Morwen Thistlethwaite and Jeffrey R. Weeks},
  journal={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… 

The 250 Knots with up to 10 Crossings

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

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

  • C. Adams
  • Mathematics
    The 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

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

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

The ropelength of complex knots

The ropelength of a knot is the minimum contour length of a tube of unit radius that traces out the knot in three dimensional space without self-overlap, colloquially the minimum amount of rope

Sampling Lissajous and Fourier Knots

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

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

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

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



IX.— Alternate ± Knots of Order Eleven

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

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

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

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

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

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

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

  • T. Kirkman
  • Art
    Transactions 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

  • C. N. Little
  • Mathematics
    Transactions 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