Discussion on colorability-of-the-pretzels
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marg...@math.utah.edu
Apr 10, 2008, 12:41:46 PM4/10/08
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Onye,
It is probably good if we think about links in general, and not only
knots (remember, the number of components of a (p,q,r) pretzel link is
gcd(p,q,r). I think maybe a good place to start here is with all
(p,q,r) pretzel links where p, q, and r are all multiples of 3. Can
we solve the problem of 3-colorability in this case?
Dan
It is probably good if we think about links in general, and not only
knots (remember, the number of components of a (p,q,r) pretzel link is
gcd(p,q,r). I think maybe a good place to start here is with all
(p,q,r) pretzel links where p, q, and r are all multiples of 3. Can
we solve the problem of 3-colorability in this case?
Dan
bch...@comcast.net
Apr 10, 2008, 5:22:12 PM4/10/08
to knot theory reu
A thought would be to try a constructive argument. For instance by
adding a crossing to one of the bands, you get one more arc. So maybe
by finding one pqr knot that is 3 colorable, you could try looking at
a p+1,q+1,r+1 knot and see if that is also 3 colorable and so on. If
that works, it could be fairly simple to generalize.
adding a crossing to one of the bands, you get one more arc. So maybe
by finding one pqr knot that is 3 colorable, you could try looking at
a p+1,q+1,r+1 knot and see if that is also 3 colorable and so on. If
that works, it could be fairly simple to generalize.
Charlotte
Apr 11, 2008, 3:58:29 PM4/11/08
to knot theory reu
In homework problem 5.2.4, I showed a labeling for the (3,3,3) Pretzel
knot. It is easy to see from there that this knot is tri-colorable,
so you may like to look at that example.
Staying at the (3,3,3) level, you could ask if the (-3,3,3) knot is
colorable. I think this is the only case you would need to check
because it should be the same as the mirror image of the (3,-3,-3)
knot.
knot. It is easy to see from there that this knot is tri-colorable,
so you may like to look at that example.
Staying at the (3,3,3) level, you could ask if the (-3,3,3) knot is
colorable. I think this is the only case you would need to check
because it should be the same as the mirror image of the (3,-3,-3)
knot.
Onyebuchi Samuel Okoro
Apr 11, 2008, 6:15:56 PM4/11/08
to knot theory reu
I believe some pretzels are tri or mod 3 colorable. So I agree with you Charlotte. The problem is that only a few pretzels are tri-colorable. It is a fact that all pretzels are colorable. But part of my project is to stipulate an algorithm by which we may actually look at a pretzel and determine its mod p colorobility. Dan wants me to extend it to links as well. Unlike the knots, some links are mod 2 colorable. So my project is really quite not as simplistic as it may appear superficially. It's going to be interesting. I believe the number of crossings in each of the three domains of the pretzel knot play an important part in determining the colorobility of the pretzels. Just how this comes to play is the essence of this project. Thanks for your insight though.
From: knotth...@googlegroups.com on behalf of Charlotte
Sent: Fri 4/11/2008 1:58 PM
To: knot theory reu
Subject: Discussion on colorability-of-the-pretzels
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