> It is a sixties thing: make love, knot war

That's brilliant!

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Mike Williams wrote:

> I was wondering what the next simplest shape similar to a moebius trefoil
> would be. It's fairly easy to make a 5-lobed knot similar to the 3-lobed
> trefoil, but my attempts to produce something with four lobes tended to
> produce structures that consisted of two loops with various degrees of
> linking between them. Then I happened to create this thing. I'm not sure if
> it's a knot or not.
> 
> 

Well, if it's the same as the first knot, below, and I think it is, then 
it's a "3,2 Torus Knot" (which I generated using KnotPlot -- 
http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html )

In which case, using KnotPlot, I deformed it (continuously, so it should 
be topologically equivalent) to the second knot, below, which looks 
somewhat familiar ;-)

Dave Matthews

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Attachments:
Download 'knotview01.jpg' (12 KB) Download 'knotview02.jpg' (15 KB)

Preview of image 'knotview01.jpg'

Preview of image 'knotview02.jpg'

> Well, if it's the same as the first knot, below, and I think it is, then 
> it's a "3,2 Torus Knot" (which I generated using KnotPlot -- 
> http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html )
> 
> In which case, using KnotPlot, I deformed it (continuously, so it should 
> be topologically equivalent) to the second knot, below, which looks 
> somewhat familiar ;-)

OK, now I'm puzzled...

The first image does indeed appear to be the same knot - as far as I can 
tell. And yet, you claim it's topologically equivilent to the trefoil 
knot. And yet... I computed the Jones polynomial for it, and it's 
different to either of the trefoil knots...

Well, one of us has fluffed up somewhere :-S Since the procedure you 
undertook is simpler, I suspect it was me :-$

Andrew.

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Andrew the Orchid wrote:

>> Well, if it's the same as the first knot, below, and I think it is, 
>> then it's a "3,2 Torus Knot" (which I generated using KnotPlot -- 
>> http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html )
>>
>> In which case, using KnotPlot, I deformed it (continuously, so it 
>> should be topologically equivalent) to the second knot, below, which 
>> looks somewhat familiar ;-)
> 
> 
> OK, now I'm puzzled...
> 
> The first image does indeed appear to be the same knot - as far as I can 
> tell. And yet, you claim it's topologically equivilent to the trefoil 
> knot. And yet... I computed the Jones polynomial for it, and it's 
> different to either of the trefoil knots...
> 
> Well, one of us has fluffed up somewhere :-S Since the procedure you 
> undertook is simpler, I suspect it was me :-$
> 
> Andrew.

I noticed that, too, but, as you noted, I didn't do anything 
mathematical, just clicked "go" and let it tighten itself up (or is it 
loosen itself up?) and it quite apparently transformed without ever 
disconnecting.  I haven't calculated a Jones polynomial since grad 
school, which is too many years ago.  I best look it up.

Dave Matthews

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Andrew the Orchid wrote:


> Well, one of us has fluffed up somewhere :-S Since the procedure you 
> undertook is simpler, I suspect it was me :-$
> 
> Andrew.

Well, again studiously avoiding doing anything resembling mathematics 
myself, I downloaded a "KnotTheory" Mathematica package ( 
http://www.math.toronto.edu/~drorbn/KAtlas/Manual/index.html ), and 
typed in Jones[TorusKnot[3,2]][q], which produced q + q^3 - q^4 (which 
is the same as for one of the Trefoil knots, and the picture the package 
gives for the (3,2) Torus knot looks like a trefoil.  See the 
attachment.  If I recall correctly (and that's a big ^if^), the 
exponents in the Jones polynomial should always work out to be positive 
or negative integers.  I suppose now I'm going to have to sit down and 
think it through again.  I'm too old for this ;-)

Dave Matthews

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Attachments:
Download 'mathematicasnap.jpg' (19 KB)

Preview of image 'mathematicasnap.jpg'

Dave Matthews wrote:

> Andrew the Orchid wrote:
> 
> 
>> Well, one of us has fluffed up somewhere :-S Since the procedure you 
>> undertook is simpler, I suspect it was me :-$
>>
>> Andrew.
> 
> 
> Well, again studiously avoiding doing anything resembling mathematics 
> myself, I downloaded a "KnotTheory" Mathematica package ( 
> http://www.math.toronto.edu/~drorbn/KAtlas/Manual/index.html ), and 
> typed in Jones[TorusKnot[3,2]][q], which produced q + q^3 - q^4 (which 
> is the same as for one of the Trefoil knots, and the picture the package 
> gives for the (3,2) Torus knot looks like a trefoil.  See the 
> attachment.  If I recall correctly (and that's a big ^if^), the 
> exponents in the Jones polynomial should always work out to be positive 
> or negative integers.  I suppose now I'm going to have to sit down and 
> think it through again.  I'm too old for this ;-)
> 
> Dave Matthews
> 
> ------------------------------------------------------------------------
> 
Now I notice that the handed-ness is reversed (he said, replying to 
himself.)  I see why:  I actually had to scale my earlier image by -1 to 
get Mike Williams' knot, and then also scaled the trefoil.  So I guess 
we should get q^-1 + q^-3 - q^-4 for the Jones polynomial of either or both.

Dave Matthews

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> Well, again studiously avoiding doing anything resembling mathematics 
> myself, I downloaded a "KnotTheory" Mathematica package ( 
> http://www.math.toronto.edu/~drorbn/KAtlas/Manual/index.html ), and 
> typed in Jones[TorusKnot[3,2]][q], which produced q + q^3 - q^4 (which 
> is the same as for one of the Trefoil knots, and the picture the package 
> gives for the (3,2) Torus knot looks like a trefoil.  See the 
> attachment.  If I recall correctly (and that's a big ^if^), the 
> exponents in the Jones polynomial should always work out to be positive 
> or negative integers.  I suppose now I'm going to have to sit down and 
> think it through again.  I'm too old for this ;-)

*You* have Mathematica? :-|

[Is envious]

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Andrew the Orchid wrote:

> *You* have Mathematica? :-|
> 
> [Is envious]

My employers have Mathematica (I teach at a community college.)  Along 
with lots of vacation time, it's one of the perks.

Dave Matthews

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>> *You* have Mathematica? :-|
>>
>> [Is envious]
> 
> 
> My employers have Mathematica (I teach at a community college.)  Along 
> with lots of vacation time, it's one of the perks.

OK.

OTOH, it's a college...

But still, one of my long-held dreams is to one day own Mathematica. 
(Although just being able to use it would also be kinda cool. Heh!)

If only it didn't cost almost as much as my car... (Well, if you're a 
student [studying in the right place], it doesn't cost that much. *sigh*)

Andrew.

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> Well, if it's the same as the first knot, below, and I think it is, then 
> it's a "3,2 Torus Knot" (which I generated using KnotPlot -- 
> http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html )
> 
> In which case, using KnotPlot, I deformed it (continuously, so it should 
> be topologically equivalent) to the second knot, below, which looks 
> somewhat familiar ;-)

Never heard of KnotPlot before.

However, in a stunning coincidence, today I did a search on Google 
[relating to knot theory], and I unded up reading a paper on drawing 
knots. And about half way through the introduction, I find that this 
paper was written by the author of something called KnotPlot...

...so I guess by the time I finish all 250+ pages, I'll know all about 
it! :-D

Andrew.

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