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.
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John Conway Knot's Don't Cancel March 1990

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Introduction to the Kauffman Bracket
Transcription
Contents
Basic concepts
Tangles
In Conway notation, the tangles are generally algebraic 2tangles. 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 NWSE 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]}
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. 329358, 1970. pdf available online
 Louis H. Kauffman, Sofia Lambropoulou: On the classification of rational tangles. Advances in Applied Mathematics, 33, No. 2 (2004), 199237. preprint available at arxiv.org.