Unique Conway notation for knots? Is the Conway notation for a knot unique? Here are two rational tangles whose closures give the trefoil knot.

However the Conway notation written for the trefoil knot is usually presented as 3 in knot tables. Could 2, -2 also be a valid Conway notation for the trefoil knot?
 A: Yes, $2\ -2$ is a valid notation for the trefoil knot (but $2,-2$ is not because comma denotes ramification, not product). For knots and links with little number of crossings a unique "nice" choice of the notation is possible. However, in general Conway notation is not unique. For example, these two diagrams represent the same link (and both projections are minimal), but in terms of Conway notation they are constructed as tangle $1$ inserted into all vertices of two different basic polyhedra.

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A: The first answer is correct -- because notation is for diagrams, of which there are many for any single knot. Conway's continued fraction solves this problem for 2-bridge knots, since equivalent notations all produce a numerically identical fraction. The same cannot be said for the two parameters of Schubert's "normal form" diagrams.
Conway's notation, when further reduced to his finite continued fraction, is thus a miracle of precision for 2-bridge knots. It's a "complete invariant" for this particular class of knots.
For example, 1/(2+(1/-2))=-3 So in this sense Conway's notation is unique for 2-bridge (a/k/a “rational”) knots.
