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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.
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Download 'KnotOrNot.jpg' (79 KB)
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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.
I missed somtehing ? s there a knot battle ? where ? wheeeere ? ;-))
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> 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.
Knot Theory - the theroy of knots and their properties - is a complex
but (IMHO) fascinating theory.
One of the topics it deals with is whether to seemingly different
arrangements of string are actually the same knot, arranged into a
different position, or whether they actually *are* different.
In this theory, the simplest knot is the "unknot" or "trivial knot" -
which is simply a closed loop of string. (Not necessarily a circle - you
can actually rearrange it into constructions which *look* like a
different knot. But if you made the thing out of physical string, you
could quickly untangle it.)
Anyway, to determine whether your knot is the unknot or whether it's a
"real knot", you could try computing one of the knot invariants on it...
(Er... or you could wait a few minutes while *I* compute some
invariants! :-S)
The idea of a knot invariant is that no matter how you rearrange the
knot, you can still (fairly) easily compute the invariant from it, and
you will still get the same answer. _However_, that doesn't mean two
totally different knots can't just happen to have the same invariant.
Long story short: if two knots have a different invariant, they are NOT
the same knot. If they have the same invariant, they MIGHT be the same knot.
Now, if you give me a moment, I'll just go Google the Jones polynomial...
Andrew.
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my two-cents : two 4-lobed knots and a 3-lobed self-intersecting knot
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Download '_tube04.jpg' (18 KB)
Download '_tube03.jpg' (16 KB)
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Preview of image '_tube04.jpg'
Preview of image '_tube03.jpg'
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OK, unless I have miscalculated, the Jones polynomial for the knot you
drew obeys
V(t)/t = t + t^(1/2) - t^(-1/2)
which, unless I'm very much mistaken, rearranges to
V(t) = t^2 + t^(3/2) - t^(1/2)
For comparism, the polynomial for the right-trefoil is
V(t) = -t^4 + t^3 + t
For the left-trefoil it is
V(t) = -t^(-4) + t^(-3) + t^(-1)
Most importantly, the Jones polynomial for the unknot is V(t) = 1. The
Jones polynomial for your knot is NOT 1 - therefore your knot is
non-trivial. QED.
Andrew.
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LightBeam wrote:
> 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.
>
>
> I missed somtehing ? s there a knot battle ? where ? wheeeere ? ;-))
It is a sixties thing: make love, knot war
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> 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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Download 'knotview02.jpg' (15 KB)
Preview of image 'knotview01.jpg'
Preview of image 'knotview02.jpg'
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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 ;-)
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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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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