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After seeing this picture of the figure eight knot:

Figure 8 knot rose limacon curve

Why isn't the figure eight knot considered a $(2,3)$-torus knot?

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Isn't the $(2,3)$ torus knot the trefoil knot? The trefoil isn't equivalent to the figure-8. I think you have to study the definition of torus knot carefully. –  Gerry Myerson Mar 26 '13 at 12:44
You could cite a theorem of Thurston classifying knots into hyperbolic, torus, and satellite. The fig-8 is hyperbolic, hence not torus. (I don't know its proof, though.) This seems to be a decent survey paper: –  Neal Mar 26 '13 at 12:58
Sorry I mean T(3,2) –  mbidwd Mar 26 '13 at 13:26
The (p,q)-torus knot is equivalent to the (q,p)-torus knot, so T(3,2) is the same as T(2,3) and is still the trefoil. –  wilsonw Jun 15 '13 at 6:25

1 Answer 1

The $(3,2)$ torus knot is also known as the "trefoil" knot. It admits a tricoloring, while the figure eight knot (that you drew) does not. Here is a reference:

To answer your question directly: The diagram you drew certainly looks like it lies on the standard two-torus in $\mathbb{R}^3$. But stare at one of four arcs that connect an outer crossing with an inner crossing. The crossing pattern at the ends of the arc prevent it from lying in the two-torus. At one of the ends, it will have to go into (or out of, respectively) the two-torus, to avoid crashing into another arc.

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