Talk:Trefoil knot

Untitled

Why has somebody put a picture of the Figure 8 knot as a pretzel?... — Preceding unsigned comment added by 139.124.3.100 (talk) 15:15, 18 March 2013 (UTC)

I have removed the offending image. --Gallusgallus (talk) 16:47, 15 January 2014 (UTC)

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

This is very difficult to understand, especially for a non-mathematician. For example, saying "and is the closure of the braid σ1³" doesn't help much. σ1³? Really. --KirbyManiac (talk) 12:13, 6 July 2008 (UTC)

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)

Number theory

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)

Bott passed away a couple years ago...so you couldn't be him! --C S (Talk) 22:14, 22 March 2007 (UTC)

unique?

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)

Error

The section Nontriviality contains this sentence:

"Mathematically, this means that a trefoil knot is not isotopic to the unknot."

The word "isotopic" (without further qualification) is an equivalence relation on topological embeddings of one space into another.

With this meaning, all knots are isotopic, simply because a knot can be "pulled tight" so that it disappears, via an isotopy.

This is a mistake in the article. 2601:200:C000:1A0:52D:8748:7384:FB5A (talk) 18:05, 26 October 2022 (UTC)

False statement

The section Descriptions, after presenting equations for a trefoil knot, makes this claim:

"Any continuous deformation of the curve above is also considered a trefoil knot."

This is simply not true, because as with any knot, a merely continuous deformation can pull the knot tight so that it disappears.

I hope someone knowledgeable about knot theory can fix this mistake.

Missing word?

The section Classification contains this sentence:

"Since the knot complement is also a Seifert fibred with boundary, it has a horizontal incompressible surface—this is also the fiber of the Milnor map."

The sentence is missing a noun after the adjectival phrase "Seifert fibred".

Perhaps the noun should be "3-manifold" ?

I hope someone knowledgeable about this subject can fix this omission.