Conway notation (knot theory)
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In knot theory, Conway notation, invented by John Horton Conway, is a way of describing knots that allows one to see many of a knots properties from it. It composes a knot using certain operations on tangles to construct it.
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[edit] Basic concepts
[edit] 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.
[edit] Operations on tangles
If a tangle, a, is reflected on the NW-SE line, it is denoted by -a. (Note that this is different than 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.
[edit] Advanced concepts
Rational tangles are equivalent iff their fractions are equal. [2] A number before an asterisk *, denotes the polyhedron number, multiple asterisks indicate that multiple polyhedra of that number exist. [3]
