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.

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.

he following seems to be attempting to describe only integer or 1/n rational tanglesTangles 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'', 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.

See also

*Conway knot
*Dowker notation
*Alexander–Briggs notation
*Gauss notation

References

Further reading

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{{Knot theory, state=collapsed

Knot theory
John Horton Conway