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Knots, Braids, andthe Vogel Algorithm
Dan Roozemond31 May 2007
12
/ department of mathematics and computer science
Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Contents
I KnotsI BraidsI Seifertview
I The Vogel AlgorithmI Knotweaver
I Conclusion
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Knots
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Representing a Knot: The Gauss Code
Gauss Code: -1, +2, -3, +4, +5, +1, -2, +6, +7, +3, -4, -7, -6, -5
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Representing a Knot: The Gauss Code
Gauss Code: -1, +2, -3, +4, +5, +1, -2, +6, +7, +3, -4, -7, -6, -5
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Representing a Knot: Oriented Gauss Code
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Representing a Knot: Oriented Gauss Code
Oriented Gauss Code:-1, +2, -3, +4, +5, +1, -2, +6, +7, +3, -4, -7, -6, -5 / - - - - + - +
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Representing a Knot: Oriented Gauss Code
Oriented Gauss Code:-1, +2, -3, +4, +5, +1, -2, +6, +7, +3, -4, -7, -6, -5 / - - - - + - +
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Seifert Circles
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Braids
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Braids
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Seifertview[vanWijk, 2005]
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Seifertview[vanWijk, 2005]
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Knots↔ Braids
I Seifertview uses braids
I Knots and braids are essentially the same things!I Braidgroups, Artin groups, Hecke algebras, etc.
So we want to convert between knots and braids!
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/ department of mathematics and computer science
Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Knots↔ Braids
I Seifertview uses braidsI Knots and braids are essentially the same things!
I Braidgroups, Artin groups, Hecke algebras, etc.
So we want to convert between knots and braids!
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/ department of mathematics and computer science
Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Knots↔ Braids
I Seifertview uses braidsI Knots and braids are essentially the same things!I Braidgroups, Artin groups, Hecke algebras, etc.
So we want to convert between knots and braids!
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/ department of mathematics and computer science
Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Knots↔ Braids
I Seifertview uses braidsI Knots and braids are essentially the same things!I Braidgroups, Artin groups, Hecke algebras, etc.
So we want to convert between knots and braids!
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/ department of mathematics and computer science
Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Knots↔ Braids
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Knots↔ Braids
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Braids to Knots
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Vogel Algorithm
Vogel, 1990A knot can be brought into suitable form by doing a series of Vogelmoves (Reidemeister II moves). The braid word can then easily beread off.
We have implemented this algorithm in GAP.
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Vogel Algorithm
Vogel, 1990A knot can be brought into suitable form by doing a series of Vogelmoves (Reidemeister II moves). The braid word can then easily beread off.
We have implemented this algorithm in GAP.
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Vogel Algorithm: On the Web
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Vogel Algorithm: On the Web
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Knots↔ Braids
Knotl
Oriented Gauss Codel
Web pagel
Braid wordl
Seifertview
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Knotweaver[Vos, 2007]
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Knotweaver[Vos, 2007]
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Knotweaver[Vos, 2007]
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Linking Knotweaver and Seifertview
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Linking Knotweaver and Seifertview
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/ department of mathematics and computer science
Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Linking Knotweaver and Seifertview
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/ department of mathematics and computer science
Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Linking Knotweaver and Seifertview
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
The Vogel Algorithm: A bit more
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/ department of mathematics and computer science
Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
The Vogel Algorithm: A bit more
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/ department of mathematics and computer science
Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
The Vogel Algorithm: A bit more
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/ department of mathematics and computer science
Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
The Vogel Algorithm: A bit more
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/ department of mathematics and computer science
Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
The Vogel Algorithm: A bit more
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Conclusion
I Implementation of the Vogel algorithmI Link between Knotweaver and Seifertview
I Future options:I Smoother connectionI Multiple componentsI Visualize Vogel algorithm in Knotweaver
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Conclusion
I Implementation of the Vogel algorithmI Link between Knotweaver and SeifertviewI Future options:
I Smoother connectionI Multiple componentsI Visualize Vogel algorithm in Knotweaver
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Knots Braids Seifertview The Vogel Algorithm Knotweaver Conclusion
Questions?
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![SOFIA LAMBROPOULOU arXiv:1811.11701v1 [math.GT] 28 Nov … · DIAGRAMMATIC REPRESENTATIONS OF KNOTS AND LINKS AS CLOSED BRAIDS SOFIA LAMBROPOULOU 1. Introduction A way to study knots](https://img.pdfslide.us/doc/110x75/5ec0651b5af82d746402eed0/sofia-lambropoulou-arxiv181111701v1-mathgt-28-nov-diagrammatic-representations.jpg)
