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Science > Physics |
| User: |
"" |
| Date: |
13 May 2007 03:00:34 AM |
| Object: |
Looking for information on finite-length braids |
I am looking for papers on "finite" three-stranded braids.
The objects I mean are the following:
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There are three strands that are glued together at each end.
The ascii drawing only shows the simplest configuration of the
trivial
braid. Obviously, there are many such objects. Some of them can
be transformed into each other, some cannot.
Where can one read more about them? Thank you for any help!
Frank
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| User: "Hero" |
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| Title: Re: Looking for information on finite-length braids |
13 May 2007 05:01:54 AM |
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On 13 Mai, 09:00, wrote:
I am looking for papers on "finite" three-stranded braids.
The objects I mean are the following:
------
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
------
There are three strands that are glued together at each end.
The ascii drawing only shows the simplest configuration of the
trivial
braid. Obviously, there are many such objects. Some of them can
be transformed into each other, some cannot.
Where can one read more about them? Thank you for any help!
http://www.virtue.to/articles/braiding.html
http://www.dreamweaverbraiding.com/Braiding_Tips.htm
http://math.ucsb.edu/~bigelow/braids.html
http://www.math.buffalo.edu/mad/special/gilmer-gloria_HAIRSTYLES.html
http://www.ccd.rpi.edu/Eglash/csdt/african/CORNROW_CURVES/
Have fun
Hero
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| User: "tarq" |
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| Title: Re: Looking for information on finite-length braids |
13 May 2007 08:24:10 AM |
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On 13 May, 09:00, wrote:
I am looking for papers on "finite" three-stranded braids.
The objects I mean are the following:
------
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
------
There are three strands that are glued together at each end.
The ascii drawing only shows the simplest configuration of the
trivial
braid. Obviously, there are many such objects. Some of them can
be transformed into each other, some cannot.
Where can one read more about them? Thank you for any help!
Frank
Mathematically a braid may be desscribed by a square matrix in which
each row and each column contains one only element equal to 1, and
wilth all other elements equat to 0, and with,say, n equalling the
individual strand number and m equalling the strand position.
Multiplying by the identity matrix leaves the strand positions
unchanged.
Multiplying by a transpositions matrix, composed of 1s and 0s as the
starting matrix, will yield the transposition result.
This can again and again be multiplied by the transposition matrix (or
by a sequence of differing such matrices) to give the overall braid
pattern.
From the above, for those into cryptography, a set of near uncrackable
codes can be generated.
Tarq
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| User: "Lee Rudolph" |
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| Title: Re: Looking for information on finite-length braids |
13 May 2007 08:52:15 AM |
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tarq <tarquinx@gmail.com> writes:
On 13 May, 09:00, wrote:
I am looking for papers on "finite" three-stranded braids.
If you really want to restrict yourself to 3-braids (which are very
special), you could start with the little book _On Closed 3-Braids_
by Kunio Murasugi.
Mathematically a braid may be desscribed by a square matrix in which
each row and each column contains one only element equal to 1, and
wilth all other elements equat to 0, and with,say, n equalling the
individual strand number and m equalling the strand position.
Multiplying by the identity matrix leaves the strand positions
unchanged.
Multiplying by a transpositions matrix, composed of 1s and 0s as the
starting matrix, will yield the transposition result.
This can again and again be multiplied by the transposition matrix (or
by a sequence of differing such matrices) to give the overall braid
pattern.
You have described the symmetric group on n letters, which is a (finite)
homomorphic image of the (infinite) braid group on n strings. The
strings of a braid can cross "over" or "under" each other: your
description ignores that information.
From the above, for those into cryptography, a set of near uncrackable
codes can be generated.
Mike Anshel and others had high hopes of that, for a while, but it turns out
not be be true (for standard interpretations of "near uncrackable").
Lee Rudolph
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| User: "" |
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| Title: Re: Looking for information on finite-length braids |
14 May 2007 12:11:02 AM |
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On May 13, 3:52 pm, (Lee Rudolph) wrote:
tarq <tarqu...@gmail.com> writes:
On 13 May, 09:00, wrote:
I am looking for papers on "finite" three-stranded braids.
If you really want to restrict yourself to 3-braids (which are very
special), you could start with the little book _On Closed 3-Braids_
by Kunio Murasugi.
Lee Rudolph
Thank you - I will look for the book asap. Is there anything about the
topic of *finite* braids to be found on the internet?
Regards
Frank
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| User: "Lee Rudolph" |
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| Title: Re: Looking for information on finite-length braids |
14 May 2007 08:14:03 AM |
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writes:
On May 13, 3:52 pm, (Lee Rudolph) wrote:
tarq <tarqu...@gmail.com> writes:
On 13 May, 09:00, wrote:
I am looking for papers on "finite" three-stranded braids.
If you really want to restrict yourself to 3-braids (which are very
special), you could start with the little book _On Closed 3-Braids_
by Kunio Murasugi.
Lee Rudolph
Thank you - I will look for the book asap. Is there anything about the
topic of *finite* braids to be found on the internet?
I have no idea what you mean by "finite length" or "finite" braids.
The usual theory of braids and closed braids *is* about what
I would call "finite length" and "finite" braids. Where are
you getting your terminology from? If you have just invented it
yourself (which is fine), then what precisely do you mean by it?
You could also try to find a copy of Joan Birman's book _Braids,
Links, and Mapping Class Groups_.
Lee Rudolph
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| User: "" |
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| Title: Re: Looking for information on finite-length braids |
15 May 2007 03:20:36 PM |
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On May 14, 3:14 pm, (Lee Rudolph) wrote:
writes:
On May 13, 3:52 pm, (Lee Rudolph) wrote:
tarq <tarqu...@gmail.com> writes:
On 13 May, 09:00, wrote:
I am looking for papers on "finite" three-stranded braids.
If you really want to restrict yourself to 3-braids (which are very
special), you could start with the little book _On Closed 3-Braids_
by Kunio Murasugi.
Lee Rudolph
Thank you - I will look for the book asap. Is there anything about the
topic of *finite* braids to be found on the internet?
I have no idea what you mean by "finite length" or "finite" braids.
The usual theory of braids and closed braids *is* about what
I would call "finite length" and "finite" braids. Where are
you getting your terminology from? If you have just invented it
yourself (which is fine), then what precisely do you mean by it?
You could also try to find a copy of Joan Birman's book _Braids,
Links, and Mapping Class Groups_.
Lee Rudolph
Thanks. Yes, I do not know how to call those braids. I am looking
for braids whose ends are glued together - that is why I call them
finite.
Regards
Frank
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| User: "Lee Rudolph" |
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| Title: Re: Looking for information on finite-length braids |
15 May 2007 03:41:35 PM |
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writes:
Thanks. Yes, I do not know how to call those braids. I am looking
for braids whose ends are glued together - that is why I call them
finite.
Okay. Those are what are usually called "closed braids", and
there is a ton of information about them; start with Birman's
book. There is also a good chapter about them in _Knots and
Links_ by Dale Rolfsen. Almost any book on knot theory will
discuss them, in fact; it's an old theorem of James Alexander
that every knot and link (as defined in topology--you may
have other meanings, or none, for the words, but once you
get Rolfsen you'll quickly learn the lingo) can be represented
by a closed braid (in fact, many many different ones).
So, now that I know what you're interested in, why are *you*
interested? (I know why I am.) Presumably for physical
reasons--which ones? (There are known applications to various
physical and chemical phenomena, some more reasonable than
others. Maybe your application is already known; maybe it's
new. Tell us!)
Lee Rudolph
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| User: "" |
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| Title: Re: Looking for information on finite-length braids |
18 May 2007 01:16:26 AM |
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On May 15, 10:41 pm, (Lee Rudolph) wrote:
frank_k_shel...@yahoo.co.uk writes:
Thanks. Yes, I do not know how to call those braids. I am looking
for braids whose ends are glued together - that is why I call them
finite.
Okay. Those are what are usually called "closed braids", and
there is a ton of information about them; start with Birman's
book.
Lee, are you sure they are the same? If the ends of the strands are
glued
together on each end, still the two glued blobs are separate.
And they can be moved
in ways that are not possible if the two blobs are glued to each
other.
I suspect that closed braids are braids where the ends on one side
are connected to the ends on the other side. This is not what I mean
(sorry if I explained this incorrectly.)
Or do I misunderstand something?
Frank
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| User: "Lee Rudolph" |
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| Title: Re: Looking for information on finite-length braids |
18 May 2007 05:32:05 AM |
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writes:
On May 15, 10:41 pm, (Lee Rudolph) wrote:
frank_k_shel...@yahoo.co.uk writes:
Thanks. Yes, I do not know how to call those braids. I am looking
for braids whose ends are glued together - that is why I call them
finite.
Okay. Those are what are usually called "closed braids", and
there is a ton of information about them; start with Birman's
book.
Lee, are you sure they are the same? If the ends of the strands are
glued
together on each end, still the two glued blobs are separate.
And they can be moved
in ways that are not possible if the two blobs are glued to each
other.
I suspect that closed braids are braids where the ends on one side
are connected to the ends on the other side. This is not what I mean
(sorry if I explained this incorrectly.)
AHA. I misunderstood. Do you want *all* the ends at the top glued
together into one point, and *all* the ends at the bottom glued
together into another pont? Or all ends glued together into one
point? And in either case, how rigid is the glue (this is a serious
question)? The general area of this question is "singular braid
theory"; it's fairly new and I haven't kept up with it at all.
I'll be off-line for a couple of days, but then I'll be at a braid
theory conference, so keep posting and maybe I can find someone who
knows what you need to know.
Lee Rudolph
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| User: "" |
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| Title: Re: Looking for information on finite-length braids |
19 May 2007 07:53:21 AM |
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On May 18, 12:32 pm, (Lee Rudolph) wrote:
AHA. I misunderstood. Do you want *all* the ends at the top glued
together into one point, and *all* the ends at the bottom glued
together into another pont?
Yes.
Or all ends glued together into one point?
No.
And in either case, how rigid is the glue (this is a serious
question)?
The glue should keep the ends from untangling.
So fully rigid is ok.
The general area of this question is "singular braid
theory"; it's fairly new and I haven't kept up with it at all.
Using the remark form another post, this means that
boy scout tricks are more advanced than modern mathematics?
(Just joking.)
Frank
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| User: "" |
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| Title: Re: Looking for information on finite-length braids |
19 May 2007 06:58:10 AM |
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In article <1179579201.058925.57800@n59g2000hsh.googlegroups.com>,
wrote:
On May 18, 12:32 pm, (Lee Rudolph) wrote:
AHA. I misunderstood. Do you want *all* the ends at the top glued
together into one point, and *all* the ends at the bottom glued
together into another pont?
Yes.
I'm trying to figure out how to do this plus keep an open loop
between the braided part and the "point".
Or all ends glued together into one point?
No.
Two ends, right?
And in either case, how rigid is the glue (this is a serious
question)?
The glue should keep the ends from untangling.
So fully rigid is ok.
But glue peels and doesn't last.
<snip>
/BAH
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| User: "Ben newsam" |
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| Title: Re: Looking for information on finite-length braids |
18 May 2007 08:04:51 AM |
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On 17 May 2007 23:16:26 -0700, wrote:
I suspect that closed braids are braids where the ends on one side
are connected to the ends on the other side. This is not what I mean
(sorry if I explained this incorrectly.)
Or do I misunderstand something?
I think that a closed braid is one where the ends are tied or fixed
together. Any (well OK, some) Boy Scout knows how to make a "woggle"
by putting two slits in a strip of leather, and manipulating the strip
by passing the ends through the slits and so on, ending up with a
braid. The braid can then be formed into a ring and fixed with a stud.
The same trick can be done with a piece of paper. Try it.
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