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Knotted Periodic Orbits in Flows
Michael Sullivan
Southern Illinois University
March 2018
Harris-Stowe University
mikesullivan (at) math (dot) siu (dot) edu
http://www.math.siu.edu/sullivan
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Knots and Links
Notation.
• Rn is the Euclidean space of dimension n.
• S1 = {(x , y) ∈ R2 | x2 + y2 = 1}.• S3 = {(w , x , y , z) ∈ R4 |w2 + x2 + y2 + z2 = 1} ≈ R3 ∪{∞}.
Definition. A knot is a smooth one-to-one map K of S1 intoR3 or S3.
K : S1 → R3 or S3
Two knots are equivalent (have the same knot-type) if one canbe smoothly deformed into the other.
![Page 3: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/3.jpg)
Knots and LinksNotation.
• Rn is the Euclidean space of dimension n.
• S1 = {(x , y) ∈ R2 | x2 + y2 = 1}.• S3 = {(w , x , y , z) ∈ R4 |w2 + x2 + y2 + z2 = 1} ≈ R3 ∪{∞}.
Definition. A knot is a smooth one-to-one map K of S1 intoR3 or S3.
K : S1 → R3 or S3
Two knots are equivalent (have the same knot-type) if one canbe smoothly deformed into the other.
![Page 4: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/4.jpg)
Knots and LinksNotation.
• Rn is the Euclidean space of dimension n.
• S1 = {(x , y) ∈ R2 | x2 + y2 = 1}.• S3 = {(w , x , y , z) ∈ R4 |w2 + x2 + y2 + z2 = 1} ≈ R3 ∪{∞}.
Definition. A knot is a smooth one-to-one map K of S1 intoR3 or S3.
K : S1 → R3 or S3
Two knots are equivalent (have the same knot-type) if one canbe smoothly deformed into the other.
![Page 5: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/5.jpg)
Knots and LinksNotation.
• Rn is the Euclidean space of dimension n.
• S1 = {(x , y) ∈ R2 | x2 + y2 = 1}.• S3 = {(w , x , y , z) ∈ R4 |w2 + x2 + y2 + z2 = 1} ≈ R3 ∪{∞}.
Definition. A knot is a smooth one-to-one map K of S1 intoR3 or S3.
K : S1 → R3 or S3
Two knots are equivalent (have the same knot-type) if one canbe smoothly deformed into the other.
![Page 6: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/6.jpg)
Knots and LinksNotation.
• Rn is the Euclidean space of dimension n.
• S1 = {(x , y) ∈ R2 | x2 + y2 = 1}.• S3 = {(w , x , y , z) ∈ R4 |w2 + x2 + y2 + z2 = 1} ≈ R3 ∪{∞}.
Definition. A knot is a smooth one-to-one map K of S1 intoR3 or S3.
K : S1 → R3 or S3
Two knots are equivalent (have the same knot-type) if one canbe smoothly deformed into the other.
![Page 7: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/7.jpg)
Knots can be Given an Orientation
![Page 8: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/8.jpg)
Knots can be Given an Orientation
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Knots and Links
Definition. A link is a smooth one-to-one map L of a disjointunion of a finite number of copies of S1 into R3 or S3.
L : S1 × {1} ∪ · · · ∪ S1 × {n} → R3 or S3
![Page 10: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/10.jpg)
Knots and Links
Definition. A link is a smooth one-to-one map L of a disjointunion of a finite number of copies of S1 into R3 or S3.
L : S1 × {1} ∪ · · · ∪ S1 × {n} → R3 or S3
![Page 11: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/11.jpg)
Knots and Links
Definition. A link is a smooth one-to-one map L of a disjointunion of a finite number of copies of S1 into R3 or S3.
L : S1 × {1} ∪ · · · ∪ S1 × {n} → R3 or S3
![Page 12: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/12.jpg)
There are Two Crossing Types
Positive Negative
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Linking Number
Given an oriented link of two components the linking number isone half the sum of the crossing signs for each crossing of the twocomponent knots.
Linking Number = −1.Linking Number = 0.
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Linking Number
Given an oriented link of two components the linking number isone half the sum of the crossing signs for each crossing of the twocomponent knots.
Linking Number = −1.Linking Number = 0.
![Page 15: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/15.jpg)
Linking Number
Given an oriented link of two components the linking number isone half the sum of the crossing signs for each crossing of the twocomponent knots.
Linking Number = −1.Linking Number = 0.
![Page 16: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/16.jpg)
Linking
Two knots are said to be unlinked is they can be smoothlydeformed so that one component is inside a ball while the other isoutside. Otherwise they are linked.
If two knots are unlinked, their linking number is zero. Theconverse is false.
Whitehead link.
![Page 17: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/17.jpg)
Linking
Two knots are said to be unlinked is they can be smoothlydeformed so that one component is inside a ball while the other isoutside. Otherwise they are linked.
If two knots are unlinked, their linking number is zero. Theconverse is false.
Whitehead link.
![Page 18: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/18.jpg)
Linking
Two knots are said to be unlinked is they can be smoothlydeformed so that one component is inside a ball while the other isoutside. Otherwise they are linked.
If two knots are unlinked, their linking number is zero. Theconverse is false.
Whitehead link.
![Page 19: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/19.jpg)
Linking
Two knots are said to be unlinked is they can be smoothlydeformed so that one component is inside a ball while the other isoutside. Otherwise they are linked.
If two knots are unlinked, their linking number is zero. Theconverse is false.
Whitehead link.
![Page 20: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/20.jpg)
A Knot in a Flow
x = x + y − x(x2 + y2)
y = −x + y − y(x2 + y2)
Source: Boyce & DiPrima, 10th edition, page 566.
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A Knot in a Flow
x = x + y − x(x2 + y2)
y = −x + y − y(x2 + y2)
Source: Boyce & DiPrima, 10th edition, page 566.
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A Knot in a Flow
x = x + y − x(x2 + y2)
y = −x + y − y(x2 + y2)
Source: Boyce & DiPrima, 10th edition, page 566.
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A Flow with Lots of Knots
Lorenz Equations.
x = σ(y − x)
y = rx − y − xz
z = −bz + xy
σ = 10 r = 28 b = 8/3
![Page 24: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/24.jpg)
A Flow with Lots of KnotsLorenz Equations.
x = σ(y − x)
y = rx − y − xz
z = −bz + xy
σ = 10 r = 28 b = 8/3
![Page 25: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/25.jpg)
A Flow with Lots of KnotsLorenz Equations.
x = σ(y − x)
y = rx − y − xz
z = −bz + xy
σ = 10 r = 28 b = 8/3
![Page 26: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/26.jpg)
A Template for Lorenz Orbits
![Page 27: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/27.jpg)
A Template for Lorenz Knots
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A Template for Lorenz Knots
![Page 29: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/29.jpg)
How many different Lorenz knots are there?
Theorem. There are infinitely many distinct Lorenz knots. [Franks& Williams, 1983]
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How many different Lorenz knots are there?
Theorem. There are infinitely many distinct Lorenz knots. [Franks& Williams, 1983]
![Page 31: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/31.jpg)
Lorenz Knots are Prime
This was proven by Williams in 1984. What does it mean?
Figure−8PrimePrime
Trefoil
![Page 32: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/32.jpg)
Lorenz Knots are PrimeThis was proven by Williams in 1984. What does it mean?
Figure−8PrimePrime
Trefoil
![Page 33: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/33.jpg)
Lorenz Knots are PrimeThis was proven by Williams in 1984. What does it mean?
Figure−8PrimePrime
Trefoil
![Page 34: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/34.jpg)
Lorenz Knots are PrimeThis was proven by Williams in 1984. What does it mean?
Figure−8Trefoil #
Composite
![Page 35: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/35.jpg)
Any two interior Lorenz knots are Linked
Interior means we exclude the two unknots in the boundary of theLorenz template.
Proof: Any pair of knots in the interior of the Lorenz templatemust cross and all the crossings in the Lorenz template are of thesame sign. Thus, the linking number will be nonzero.
![Page 36: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/36.jpg)
Any two interior Lorenz knots are Linked
Interior means we exclude the two unknots in the boundary of theLorenz template.
Proof: Any pair of knots in the interior of the Lorenz templatemust cross and all the crossings in the Lorenz template are of thesame sign. Thus, the linking number will be nonzero.
![Page 37: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/37.jpg)
Any two interior Lorenz knots are Linked
Interior means we exclude the two unknots in the boundary of theLorenz template.
Proof: Any pair of knots in the interior of the Lorenz templatemust cross and all the crossings in the Lorenz template are of thesame sign. Thus, the linking number will be nonzero.
![Page 38: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/38.jpg)
Torus Knots
A torus is the surface of a donut shape. More technically
T 2 = S1 × S1.
A torus knot is a knot that can be embedded into a torus.
Image source: http://makerhome.blogspot.com/2014/01/day-150-trefoil-torus-knots.html
![Page 39: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/39.jpg)
Torus KnotsA torus is the surface of a donut shape. More technically
T 2 = S1 × S1.
A torus knot is a knot that can be embedded into a torus.
Image source: http://makerhome.blogspot.com/2014/01/day-150-trefoil-torus-knots.html
![Page 40: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/40.jpg)
Torus KnotsA torus is the surface of a donut shape. More technically
T 2 = S1 × S1.
A torus knot is a knot that can be embedded into a torus.
Image source: http://makerhome.blogspot.com/2014/01/day-150-trefoil-torus-knots.html
![Page 41: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/41.jpg)
Torus KnotsA torus is the surface of a donut shape. More technically
T 2 = S1 × S1.
A torus knot is a knot that can be embedded into a torus.
Image source: http://makerhome.blogspot.com/2014/01/day-150-trefoil-torus-knots.html
![Page 42: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/42.jpg)
Torus Knots are Lorenz Knots
Proof!
Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan.
![Page 43: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/43.jpg)
Torus Knots are Lorenz Knots
Proof!
Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan.
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Lorenz-like Templates
If we put some number of twists in each band we get aLorenz-like template. They are denoted by L(m, n) where m andn give the number of half twists in the respective bands as shown.
![Page 45: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/45.jpg)
Lorenz-like TemplatesIf we put some number of twists in each band we get aLorenz-like template. They are denoted by L(m, n) where m andn give the number of half twists in the respective bands as shown.
![Page 46: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/46.jpg)
Lorenz-like TemplatesIf we put some number of twists in each band we get aLorenz-like template. They are denoted by L(m, n) where m andn give the number of half twists in the respective bands as shown.
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Template Relationships
Theorem. Regarding L(m, n) as the set of all knots in L(m, n) wehave
· · · ⊃ L(0,−4) ⊃ L(0,−2) ⊃ L(0, 0) ⊃ L(0, 2) ⊃ · · ·
and
· · · ⊃ L(0,−3) ⊃ L(0,−1) ⊃ L(0, 1) ⊃ L(0, 3) ⊃ · · ·
![Page 48: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/48.jpg)
Template Relationships
Theorem. Regarding L(m, n) as the set of all knots in L(m, n) wehave
· · · ⊃ L(0,−4) ⊃ L(0,−2) ⊃ L(0, 0) ⊃ L(0, 2) ⊃ · · ·
and
· · · ⊃ L(0,−3) ⊃ L(0,−1) ⊃ L(0, 1) ⊃ L(0, 3) ⊃ · · ·
![Page 49: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/49.jpg)
Proof!
![Page 50: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/50.jpg)
Template Relationships
Theorem. L(0,−4) ⊂ L(0,−1).Proof!
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Template RelationshipsTheorem. L(0,−4) ⊂ L(0,−1).
Proof!
![Page 52: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/52.jpg)
Template RelationshipsTheorem. L(0,−4) ⊂ L(0,−1).Proof!
![Page 53: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/53.jpg)
Universal Templates
A template is said to be universal if it contains all knots.
Birman and Williams had conjectured these did not exist.
Theorem. L(0,−1) contains all knots. [Ghrist, 1996]
The proof is hard.
![Page 54: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/54.jpg)
Universal Templates
A template is said to be universal if it contains all knots.
Birman and Williams had conjectured these did not exist.
Theorem. L(0,−1) contains all knots. [Ghrist, 1996]
The proof is hard.
![Page 55: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/55.jpg)
Universal Templates
A template is said to be universal if it contains all knots.
Birman and Williams had conjectured these did not exist.
Theorem. L(0,−1) contains all knots. [Ghrist, 1996]
The proof is hard.
![Page 56: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/56.jpg)
Universal Templates
A template is said to be universal if it contains all knots.
Birman and Williams had conjectured these did not exist.
Theorem. L(0,−1) contains all knots. [Ghrist, 1996]
The proof is hard.
![Page 57: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/57.jpg)
Universal Templates
A template is said to be universal if it contains all knots.
Birman and Williams had conjectured these did not exist.
Theorem. L(0,−1) contains all knots. [Ghrist, 1996]
The proof is hard.
![Page 58: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/58.jpg)
An ODE with all knots
Theorem. There exists an open set of parameters β ∈ [6.5, 10.5]for which periodic solutions to the differential equation
x = 7|y − φ(x)|y = x − y + z
z = −βy
whereφ(x) = 2x/7− 3 [|x + 1| − |x − 1|] /14,
are modeled by L(0,−1).[Ghrist & Holmes, 1996]
![Page 59: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/59.jpg)
An ODE with all knots
Theorem. There exists an open set of parameters β ∈ [6.5, 10.5]for which periodic solutions to the differential equation
x = 7|y − φ(x)|y = x − y + z
z = −βy
whereφ(x) = 2x/7− 3 [|x + 1| − |x − 1|] /14,
are modeled by L(0,−1).[Ghrist & Holmes, 1996]
![Page 60: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/60.jpg)
Robinson’s Attractor and Templates
x(t) = y
y(t) = x − 2x3 − 0.71y + 1.644x2y + yz
z(t) = −0.7061z + 0.1x2
![Page 61: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/61.jpg)
Robinson’s Attractor and Templates
x(t) = y
y(t) = x − 2x3 − 0.71y + 1.644x2y + yz
z(t) = −0.7061z + 0.1x2
![Page 62: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/62.jpg)
Robinson’s Attractor and Templates
x(t) = y
y(t) = x − 2x3 − 0.71y + 1.644x2y + yz
z(t) = −0.7061z + 0.1x2
![Page 63: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/63.jpg)
Robinson’s Attractor and Templates
x(t) = y
y(t) = x − 2x3 − 0.71y + 1.644x2y + yz
z(t) = −0.7061z + 0.1x2
![Page 64: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/64.jpg)
Robinson’s Attractor and Templates
x(t) = y
y(t) = x − 2x3 − 0.71y + 1.644x2y + yz
z(t) = −0.7061z + 0.1x2
![Page 65: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/65.jpg)
An Alternative form of L(−1,−1)
![Page 66: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/66.jpg)
An Alternative form of L(−1,−1)
![Page 67: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/67.jpg)
Current work
It is not universal because ...
L(−1,−1) knots can be presented with all crossing the same sign.
My former PhD student Ghazwan Alhashimi has shown that allL(−1,−1) knots are fibered, but this is rather technical.
L(−1,−1) contains composite knots. It is now known what themaximum number of prime factors is three.
![Page 68: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/68.jpg)
Current work
It is not universal because ...
L(−1,−1) knots can be presented with all crossing the same sign.
My former PhD student Ghazwan Alhashimi has shown that allL(−1,−1) knots are fibered, but this is rather technical.
L(−1,−1) contains composite knots. It is now known what themaximum number of prime factors is three.
![Page 69: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/69.jpg)
Current work
It is not universal because ...
L(−1,−1) knots can be presented with all crossing the same sign.
My former PhD student Ghazwan Alhashimi has shown that allL(−1,−1) knots are fibered, but this is rather technical.
L(−1,−1) contains composite knots. It is now known what themaximum number of prime factors is three.
![Page 70: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/70.jpg)
Current work
It is not universal because ...
L(−1,−1) knots can be presented with all crossing the same sign.
My former PhD student Ghazwan Alhashimi has shown that allL(−1,−1) knots are fibered, but this is rather technical.
L(−1,−1) contains composite knots. It is now known what themaximum number of prime factors is three.
![Page 71: Knotted Periodic Orbits in Flowsgalileo.math.siu.edu/Talks/HS2018.pdf · Image source: Knots and Links in Three-Dimensional Flows, by Ghrist, Holmes, Sullivan. Torus Knots are Lorenz](https://reader033.vdocument.in/reader033/viewer/2022051207/6034bd3b99d43732521bdc97/html5/thumbnails/71.jpg)
Current work
It is not universal because ...
L(−1,−1) knots can be presented with all crossing the same sign.
My former PhD student Ghazwan Alhashimi has shown that allL(−1,−1) knots are fibered, but this is rather technical.
L(−1,−1) contains composite knots. It is now known what themaximum number of prime factors is three.