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# Conway notation (knot theory)

The full set of fundamental transformations and operations on 2-tangles, alongside the elementary tangles 0, ∞, ±1 and ±2.
The trefoil knot has Conway notation [3].

In knot theory, Conway notation, invented by John Horton Conway, is a way of describing knots that makes many of their properties clear. It composes a knot using certain operations on tangles to construct it.

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• John Conway Knot's Don't Cancel March 1990
• Dr. Radmila Sazdanovic - To cut or to knot
• John Conway on "The remarkable history of Pascal's hexagrammum mysticum"

## Basic concepts

### Tangles

In Conway notation, the tangles are generally algebraic 2-tangles. This means their tangle diagrams consist of 2 arcs and 4 points on the edge of the diagram; furthermore, they are built up from rational tangles using the Conway operations.

[The following seems to be attempting to describe only integer or 1/n rational tangles] Tangles consisting only of positive crossings are denoted by the number of crossings, or if there are only negative crossings it is denoted by a negative number. If the arcs are not crossed, or can be transformed to into an uncrossed position with the Reidemeister moves, it is called the 0 or ∞ tangle, depending on the orientation of the tangle.

### Operations on tangles

If a tangle, a, is reflected on the NW-SE line, it is denoted by a. (Note that this is different from a tangle with a negative number of crossings.)Tangles have three binary operations, sum, product, and ramification,[1] however all can be explained using tangle addition and negation. The tangle product, a b, is equivalent to a+b. and ramification or a,b, is equivalent to a+b.

Rational tangles are equivalent if and only if their fractions are equal. An accessible proof of this fact is given in (Kauffman and Lambropoulou 2004). A number before an asterisk, *, denotes the polyhedron number; multiple asterisks indicate that multiple polyhedra of that number exist. [2]