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.

## Contents

## 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

*. and ramification or*

^{−}a+b*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]}

## See also

## References

**^**"Conway notation",*mi.sanu.ac.rs*.**^**"Conway_Notation",*The Knot Atlas*.

## Further reading

- Conway, J. H. "An Enumeration of Knots and Links, and Some of Their Algebraic Properties." In J. Leech (editor),
*Computational Problems in Abstract Algebra.*Oxford, England. Pergamon Press, pp. 329-358, 1970. pdf available online - Louis H. Kauffman, Sofia Lambropoulou: On the classification of rational tangles. Advances in Applied Mathematics, 33, No. 2 (2004), 199-237. preprint available at arxiv.org.