[Wiskunde]PO: Knopentheorie
Moderators: ArcherBarry, Fuzzwood
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[Wiskunde]PO: Knopentheorie
kunt iemand me alsublieft uitleggen hoe de Alexander polynoom en Jones polynoom werken????
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Re: [Wiskunde]PO: Knopentheorie
heb je hier al eens gekeken: http://www.science.uva.nl/~geer/knopen.pdf, of gezocht bij Google op "Alexander polynoom"
Ik weet weinig van veel
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Re: [Wiskunde]PO: Knopentheorie
ja, die heb ik wel gelezen, en hieronder heb ik de alexander polynoom
1) Number the knot's crossings, 1 N. Prepare an N×N matrix M. (Q: does any ol' diagram do, or does it have to have minimal crossings?)
2) Walk along the knot. As you pass over crossing n, with crossing p on the left and crossing q on the right, add to the matrix:
Mnn = 1 - x
Mnp = x
Mnq = - 1
3) Fill the rest of M with zeros.
4) Drop from M any one row and any one column.
5) Take the determinant of M (this is an Alexander polynomial of the knot).
6) Normalise by dropping all the zero roots and, if the highest-degree coefficient is negative, negating.
The result is the Alexander polynomial of the knot.
Example
On a trefoil knot:
knot crossings
n p q
1 2 3
2 3 1
3 1 2
resulting in the matrix
Take the minor M23
trefoil: x2 - x + 1
of dit http://en.wikipedia.org/wiki/Alexander_polynomial
maar ik kan niet begrijp hoe ze dat doen
1) Number the knot's crossings, 1 N. Prepare an N×N matrix M. (Q: does any ol' diagram do, or does it have to have minimal crossings?)
2) Walk along the knot. As you pass over crossing n, with crossing p on the left and crossing q on the right, add to the matrix:
Mnn = 1 - x
Mnp = x
Mnq = - 1
3) Fill the rest of M with zeros.
4) Drop from M any one row and any one column.
5) Take the determinant of M (this is an Alexander polynomial of the knot).
6) Normalise by dropping all the zero roots and, if the highest-degree coefficient is negative, negating.
The result is the Alexander polynomial of the knot.
Example
On a trefoil knot:
knot crossings
n p q
1 2 3
2 3 1
3 1 2
resulting in the matrix
Take the minor M23
trefoil: x2 - x + 1
of dit http://en.wikipedia.org/wiki/Alexander_polynomial
maar ik kan niet begrijp hoe ze dat doen
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- Berichten: 4
Re: [Wiskunde]PO: Knopentheorie
Ik begrijp niet hoe ze die matrix maakt en de determinant bepaalt
Hier kun je het doornemen http://en.wikipedia.org/wiki/Talk:Knot_polynomial bij Alexander polynomial.
Ik hoop dat je mij kan helpen
Hier kun je het doornemen http://en.wikipedia.org/wiki/Talk:Knot_polynomial bij Alexander polynomial.
Ik hoop dat je mij kan helpen