Ever since
Sir William Thomson
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...
's
vortex theory, mathematicians have tried to classify and tabulate all possible
knots
A knot is a fastening in rope or interwoven lines.
Knot may also refer to:
Places
* Knot, Nancowry, a village in India
Archaeology
* Knot of Isis (tyet), symbol of welfare/life.
* Minoan snake goddess figurines#Sacral knot
Arts, entertainme ...
. As of May 2008, all
prime knot
In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be co ...
s up to 16
crossings
Crossings may refer to:
* ''Crossings'' (Buffy novel), a 2002 original novel based on the U.S. television series ''Buffy the Vampire Slayer''
* Crossings (game), a two-player abstract strategy board game invented by Robert Abbott
* ''Crossings'' ...
have been tabulated.
The major challenge of the process is that many apparently different knots may actually be different geometrical presentations of the same topological entity, and that proving or disproving
knot equivalence
In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...
is much more difficult than it at first seems.
Beginnings
In the 19th century, Sir
William Thomson made a hypothesis that the chemical elements were based upon knotted vortices in the aether. In an attempt to make a
periodic table of the elements
The periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an icon of c ...
,
P. G. Tait
Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook '' Treatise on Natural Philosophy'', which he co-wrote w ...
,
C. N. Little and others started to attempt to count all possible knots.
Because their work predated the invention of the digital computer, all work had to be done by hand.
Perko pair
In 1974, Kenneth Perko discovered a duplication in the Tait-Little tables, called the
Perko pair
In the mathematical theory of knots, the Perko pair, named after Kenneth Perko, is a pair of entries in classical knot tables that actually represent the same knot. In Dale Rolfsen's knot table, this supposed pair of distinct knots is labeled 10 ...
. Later knot tables took two approaches to resolving this: some just skipped one of the entries without renumbering, and others renumbered the later entries to remove the hole. The resulting ambiguity has continued to the present day, and has been further compounded by mistaken attempts to correct errors caused by this that were themselves incorrect. For example, Wolfram Web's Perko Pair page erroneously compares two different knots (due to the renumbering by mathematicians such as Burde and Bar-Natan).
New methods
Jim Hoste,
Jeff Weeks, and
Morwen Thistlethwaite
Morwen Bernard Thistlethwaite is a knot theorist and professor of mathematics for the University of Tennessee in Knoxville. He has made important contributions to both knot theory and Rubik's Cube group theory.
Biography
Morwen Thistlethwait ...
used computer searches to count all knots with 16 or fewer crossings. This research was performed separately using two different algorithms on different computers, lending support to the correctness of its results. Both counts found 1701936
prime knot
In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be co ...
s (including the
unknot
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...
) with up to 16 crossings.
[.] Most recently, in 2020, Benjamin Burton classified all prime knots up to 19 crossings (of which there are almost 300 million).
Starting with three crossings (the minimum for any nontrivial knot), the number of prime knots for each number of crossings is
:1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705, ...
Modern automated methods can now enumerate billions of knots in a matter of days.
See also
*
Knot theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
*
Knot (mathematics)
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of ...
*
List of prime knots
In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed here for quick comparison of their properties and varied naming schemes.
Table of prime ...
References
{{DEFAULTSORT:Knot Tabulation
Knot theory