Notation used to describe knots based on operations on tangles

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The full set of fundamental transformations and operations on 2-tangles, alongside the elementary tangles 0, ∞, ±1 and ±2.

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.

## 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 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.

## Advanced concepts[]

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]}

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