Conway notation (knot theory)

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.

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.

An accessible proof of this fact is given in (Kauffman and Lambropoulou 2004).

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].