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This looks pretty full on - Maybe a little bit keyword heavy?
- I wonder if it would be better to just link to Dror Bar-Natan's wiki rather than duplicate his efforts here? I guess this is intended to be less technical? -rb
- I added some comments to hopefully make it a little less confusing. But yeah, this article isn't written terribly well. Rybu (talk) 22:20, 6 July 2008 (UTC)
Both the presentation x^2=y^3 and the Alexander polynomial x^2-x+1 indicate this is related to the classic modular group of number theory. I've heard that the fundamental domain SL(2,R)\SO(2)/SL(2,Z) is some sort of \SO(2) congruence on the trefoil knot, or something like that, but I don't see how. I would be nice to see the details. linas 05:27, 9 November 2006 (UTC)
- The correspondence you're looking for, I think it comes from this: SL(2,R)/SL(2,Z) is diffeomorphic to the complement of the trefoil. There's a lot of proofs of this in the literature, I think the first person to observe it might be Raoul Bott. Here is an elementary argument: SO(2) acts on the left. This makes SL(2,R)/SL(2,Z) into a 3-dimensional Seifert-fibred manifold. The trefoil's complement is also a 3-dimensional Seifert fibred manifold. The trefoil complement has two singular fibres, how many does SL(2,R)/SL(2,Z) have? I think it's two: one from the tiling of the plane by squares, the other from the tiling of the plane by hexagons. Then you compare the fibre data to get the equivalence. There is a similar correspondence: consider the space of at most 3 points in the circle. This is also 3-dimensional and is seifert-fibred by the SO(2) action on the circle. The two singular fibres are the antipodal subspace and the equilateral triangle subspace, so it's also the trefoil complement... there was a note on this in the AMS Notices a few years ago. -rb (not Raoul Bott)
The article says that this is the unique knot with three crossings, but the knot and it's mirror image are not isotopic and both have three crossings. Maybe it should say unique up to chirality? —Preceding unsigned comment added by Ixionid (talk • contribs) 20:33, 7 April 2008 (UTC)