Editing Knot theory (section)

===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.