Editing Knot theory (section)

==Tabulating knots==
<!-- {{Main|knot tabulation}} more can be said about tabulation.... -->
{{see also|List of prime knots|Knot tabulation}}
[[File:Knot table.svg|thumb|upright=1.5|A table of prime knots up to seven crossings. The knots are labeled with Alexander–Briggs notation]]
Traditionally, knots have been catalogued in terms of [[crossing number (knot theory)|crossing number]].  Knot tables generally include only prime knots, and only one entry for a knot and its mirror image (even if they are different) {{Harv|Hoste|Thistlethwaite|Weeks|1998}}. The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult {{Harv|Hoste|2005|p=20}}. Tabulation efforts have succeeded in enumerating over 6 billion knots and links {{Harv|Hoste|2005|p=28}}. The sequence of the number of prime knots of a given crossing number, up to crossing number 16, is 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, {{val|46972}}, {{val|253293}}, {{val|1388705}}... {{OEIS|id=A002863}}. While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing {{Harv|Adams|2004}}.

The first knot tables by Tait, Little, and Kirkman used knot diagrams, although Tait also used a precursor to the [[Dowker notation]]. Different notations have been invented for knots which allow more efficient tabulation {{Harv|Hoste|2005}}.

The early tables attempted to list all knots of at most 10 crossings, and all alternating knots of 11 crossings {{Harv|Hoste|Thistlethwaite|Weeks|1998}}. The development of knot theory due to Alexander, Reidemeister, Seifert, and others eased the task of verification and tables of knots up to and including 9 crossings were published by Alexander–Briggs and Reidemeister in the late 1920s.

The first major verification of this work was done in the 1960s by [[John Horton Conway]], who not only developed a new notation but also the [[Alexander–Conway polynomial]] {{Harv|Conway|1970}} {{Harv|Doll|Hoste|1991}}. This verified the list of knots of at most 11 crossings and a new list of links up to 10 crossings. Conway found a number of omissions but only one duplication in the Tait–Little tables; however he missed the duplicates called the [[Perko pair]], which would only be noticed in 1974 by [[Kenneth Perko]] {{Harv|Perko|1974}}. This famous error would propagate when Dale Rolfsen added a knot table in his influential text, based on Conway's work. Conway's 1970 paper on knot theory also contains a typographical duplication on its non-alternating 11-crossing knots page and omits 4 examples — 2 previously listed in D. Lombardero's 1968 Princeton senior thesis and 2 more subsequently discovered by [[Alain Caudron]]. [see Perko (1982),  Primality of certain knots, Topology Proceedings] Less famous is the duplicate in his 10 crossing link table: 2.-2.-20.20 is the mirror of 8*-20:-20. [See Perko (2016), Historical highlights of non-cyclic knot theory, J. Knot Theory Ramifications].

In the late 1990s Hoste, Thistlethwaite, and Weeks tabulated all the knots through 16 crossings {{Harv|Hoste|Thistlethwaite|Weeks|1998}}.  In 2003 Rankin, Flint, and Schermann, tabulated the [[alternating knot]]s through 22 crossings {{Harv|Hoste|2005}}. In 2020 Burton tabulated all [[prime knot]]s with up to 19 crossings {{Harv|Burton|2020}}.

===Alexander–Briggs notation===<!--this section is linked to from locations including [[Template:Infobox knot theory]]-->
This is the most traditional notation, due to the 1927 paper of [[James Waddell Alexander II|James W. Alexander]] and [[Garland Baird Briggs|Garland B. Briggs]] and later extended by [[Dale Rolfsen]] in his knot table (see image above and [[List of prime knots]]). The notation simply organizes knots by their crossing number. One writes the crossing number with a subscript to denote its order amongst all knots with that crossing number. This order is arbitrary and so has no special significance (though in each number of crossings the [[twist knot]] comes after the [[torus knot]]). Links are written by the crossing number with a superscript to denote the number of components and a subscript to denote its order within the links with the same number of components and crossings. Thus the trefoil knot is notated 3<sub>1</sub> and the Hopf link is 2{{sup sub|2|1}}.  Alexander–Briggs names in the range 10<sub>162</sub> to 10<sub>166</sub> are ambiguous, due to the discovery of the [[Perko pair]] in [[Charles Newton Little]]'s original and subsequent knot tables, and differences in approach to correcting this error in knot tables and other publications created after this point.<ref>"[http://richardelwes.co.uk/2013/08/14/the-revenge-of-the-perko-pair/ The Revenge of the Perko Pair]", ''RichardElwes.co.uk''. Accessed February 2016. Richard Elwes points out a common mistake in describing the Perko pair.</ref>

===Dowker–Thistlethwaite notation===
{{main|Dowker–Thistlethwaite notation}}
[[File:Dowker-notation-example.svg|thumb|upright=.85|A knot diagram with crossings labelled for a Dowker sequence]]
The [[Dowker–Thistlethwaite notation]], also called the Dowker notation or code, for a knot is a finite sequence of even integers. The numbers are generated by following the knot and marking the crossings with consecutive integers. Since each crossing is visited twice, this creates a pairing of even integers with odd integers. An appropriate sign is given to indicate over and undercrossing. For example, in this figure the knot diagram has crossings labelled with the pairs (1,6) (3,&minus;12) (5,2) (7,8) (9,&minus;4) and (11,&minus;10). The Dowker–Thistlethwaite notation for this labelling is the sequence: 6, &minus;12, 2, 8, &minus;4, &minus;10. A knot diagram has more than one possible Dowker notation, and there is a well-understood ambiguity when reconstructing a knot from a Dowker–Thistlethwaite notation.

===Conway notation===
{{main|Conway notation (knot theory)}}

The [[Conway notation (knot theory)|Conway notation]] for knots and links, named after [[John Horton Conway]], is based on the theory of [[tangle (mathematics)|tangles]] {{Harv|Conway|1970}}. The advantage of this notation is that it reflects some properties of the knot or link.

The notation describes how to construct a particular link diagram of the link. Start with a ''basic polyhedron'', a 4-valent connected planar graph with no [[digon]] regions. Such a polyhedron is denoted first by the number of vertices then a number of asterisks which determine the polyhedron's position on a list of basic polyhedra. For example, 10** denotes the second 10-vertex polyhedron on Conway's list.

Each vertex then has an [[algebraic tangle]] substituted into it (each vertex is oriented so there is no arbitrary choice in substitution). Each such tangle has a notation consisting of numbers and + or &minus; signs.

An example is 1*2 &minus;3 2. The 1* denotes the only 1-vertex basic polyhedron. The 2 &minus;3 2 is a sequence describing the continued fraction associated to a [[rational tangle]]. One inserts this tangle at the vertex of the basic polyhedron&nbsp;1*.

A more complicated example is 8*3.1.2 0.1.1.1.1.1 Here again 8* refers to a basic polyhedron with 8 vertices. The periods separate the notation for each tangle.

Any link admits such a description, and it is clear this is a very compact notation even for very large crossing number. There are some further shorthands usually used. The last example is usually written 8*3:2 0, where the ones are omitted and kept the number of dots excepting the dots at the end. For an algebraic knot such as in the first example, 1* is often omitted.

Conway's pioneering paper on the subject lists up to 10-vertex basic polyhedra of which he uses to tabulate links, which have become standard for those links. For a further listing of higher vertex polyhedra, there are nonstandard choices available.

===Gauss code===
{{main|Gauss code}}
[[Gauss code]], similar to the Dowker–Thistlethwaite notation, represents a knot with a sequence of integers. However, rather than every crossing being represented by two different numbers, crossings are labeled with only one number. When the crossing is an overcrossing, a positive number is listed. At an undercrossing, a negative number. For example, the trefoil knot in Gauss code can be given as: 1,−2,3,−1,2,−3

Gauss code is limited in its ability to identify knots. This problem is partially addressed with by the [[extended Gauss code]].