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. . . . . .
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......
Energies and residues of manifoldsand configuration space of polygons
Plan of Lectures and Tutorials
Jun O’Hara (Chiba University)
June 2019
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 1 / 34
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. . . . . .
Where is Chiba?
We are here
Chiba
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 2 / 34
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. . . . . .
What does Chiba mean?
Chiba = 千葉 = thousand leaves = mille feuilles
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 3 / 34
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. . . . . .
Purpose and outline of Lectures. The space
X : a submanifold of RN ; eitherMm : a closed submanifold (∂M = ∅ and m < N)ΩN : a compact body (= the closure of the interior of Ω)
We do not consider Wm ⊂ RN with ∂W = ∅ and m < N .
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 4 / 34
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. . . . . .
Purpose and outline of Lectures. The space
X : a submanifold of RN ; eitherMm : a closed submanifold (∂M = ∅ and m < N)ΩN : a compact body (= the closure of the interior of Ω)
We do not consider Wm ⊂ RN with ∂W = ∅ and m < N .
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 4 / 34
![Page 6: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/6.jpg)
. . . . . .
Purpose and outline of Lectures. The space
X : a submanifold of RN ; eitherMm : a closed submanifold (∂M = ∅ and m < N)ΩN : a compact body (= the closure of the interior of Ω)
We do not consider Wm ⊂ RN with ∂W = ∅ and m < N .
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 4 / 34
![Page 7: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/7.jpg)
. . . . . .
Purpose and outline of Lectures. The space
X : a submanifold of RN ; either
Mm : a closed submanifold (∂M = ∅ and m < N)ΩN : a compact body (= the closure of the interior of Ω)
We do not consider Wm ⊂ RN with ∂W = ∅ and m < N .
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 4 / 34
![Page 8: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/8.jpg)
. . . . . .
The quantities
We derive two quantities for X from
∫∫X×X
|x− y|s dxdy
Energies : geometric complexity with information on global shapee.g., knot energies (cf. KnotPlot by Rob Scharein),generalized Riesz energies
Residues :
∫(local quantities),
e.g., volume (of ∂Ω), total squared curvature, Willmore functional,(Euler characteristics when dimX is small)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 5 / 34
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. . . . . .
The quantities
We derive two quantities for X from
∫∫X×X
|x− y|s dxdy
Energies : geometric complexity with information on global shapee.g., knot energies (cf. KnotPlot by Rob Scharein),generalized Riesz energies
Residues :
∫(local quantities),
e.g., volume (of ∂Ω), total squared curvature, Willmore functional,(Euler characteristics when dimX is small)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 5 / 34
![Page 10: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/10.jpg)
. . . . . .
The quantities
We derive two quantities for X from
∫∫X×X
|x− y|s dxdy
Energies : geometric complexity with information on global shapee.g., knot energies (cf. KnotPlot by Rob Scharein),generalized Riesz energies
Residues :
∫(local quantities),
e.g., volume (of ∂Ω), total squared curvature, Willmore functional,(Euler characteristics when dimX is small)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 5 / 34
![Page 11: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/11.jpg)
. . . . . .
Machinery
Metric (distance function on X ×X)
Start with I(X, s) :=
∫∫X×X
|x− y|s dxdy
I(X, s) blows up when s is small (s ≤ − dimX)
∃ Two kinds of regularizationfrom the theory of generalized functions;
Hadamard regularization (HR) andregularization via analytic continuation (AC)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 6 / 34
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. . . . . .
Machinery
Metric (distance function on X ×X)
Start with I(X, s) :=
∫∫X×X
|x− y|s dxdy
I(X, s) blows up when s is small (s ≤ − dimX)
∃ Two kinds of regularizationfrom the theory of generalized functions;
Hadamard regularization (HR) andregularization via analytic continuation (AC)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 6 / 34
![Page 13: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/13.jpg)
. . . . . .
Machinery
Metric (distance function on X ×X)
Start with I(X, s) :=
∫∫X×X
|x− y|s dxdy
I(X, s) blows up when s is small (s ≤ − dimX)
∃ Two kinds of regularizationfrom the theory of generalized functions;
Hadamard regularization (HR) andregularization via analytic continuation (AC)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 6 / 34
![Page 14: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/14.jpg)
. . . . . .
Hadamard regularization
When s is small (s ≤ − dimX),
∫∫X×X
|x− y|s dxdy blows up
on the diagonal set ∆ = (x, x) : x ∈ X.
Consider
∫∫X×X\Nε(∆)
|x− y|s dxdy (ε > 0),
expand it in a series in1
ε(a Laurent series of ε)
The constant term is called Hadamard’s finite part,
denoted by Pf.
∫∫X×X
|x− y|s dxdy
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 7 / 34
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. . . . . .
Hadamard regularization
When s is small (s ≤ − dimX),
∫∫X×X
|x− y|s dxdy blows up
on the diagonal set ∆ = (x, x) : x ∈ X.
Consider
∫∫X×X\Nε(∆)
|x− y|s dxdy (ε > 0),
expand it in a series in1
ε(a Laurent series of ε)
The constant term is called Hadamard’s finite part,
denoted by Pf.
∫∫X×X
|x− y|s dxdy
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 7 / 34
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. . . . . .
Hadamard regularization
When s is small (s ≤ − dimX),
∫∫X×X
|x− y|s dxdy blows up
on the diagonal set ∆ = (x, x) : x ∈ X.
Consider
∫∫X×X\Nε(∆)
|x− y|s dxdy (ε > 0),
expand it in a series in1
ε(a Laurent series of ε)
The constant term is called Hadamard’s finite part,
denoted by Pf.
∫∫X×X
|x− y|s dxdy
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 7 / 34
![Page 17: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/17.jpg)
. . . . . .
Regularization via analytic continuation
Consider the power s in
∫∫X×X
|x− y|s dxdy as a complex variable
(denoted by z in what follows)
C ∋ z 7→∫∫
X×X|x− y|z dxdy ∈ C as a complex function
It is holomorphic when ℜe z is big (ℜe z > −dimX)
Expand the domain by analytic continuationto obtain a meromorphic function with simple poles, BX(z),Brylinski’s beta function of X
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 8 / 34
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. . . . . .
Regularization via analytic continuation
Consider the power s in
∫∫X×X
|x− y|s dxdy as a complex variable
(denoted by z in what follows)
C ∋ z 7→∫∫
X×X|x− y|z dxdy ∈ C as a complex function
It is holomorphic when ℜe z is big (ℜe z > −dimX)
Expand the domain by analytic continuationto obtain a meromorphic function with simple poles, BX(z),Brylinski’s beta function of X
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 8 / 34
![Page 19: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/19.jpg)
. . . . . .
Regularization via analytic continuation
Consider the power s in
∫∫X×X
|x− y|s dxdy as a complex variable
(denoted by z in what follows)
C ∋ z 7→∫∫
X×X|x− y|z dxdy ∈ C as a complex function
It is holomorphic when ℜe z is big (ℜe z > −dimX)
Expand the domain by analytic continuationto obtain a meromorphic function with simple poles, BX(z),Brylinski’s beta function of X
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 8 / 34
![Page 20: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/20.jpg)
. . . . . .
Energies and residues by two regularizations
Hadamard regularization.
Laurent series p(s; ε) =
∫∫X×X\Nε(∆)
|x− y|s dxdy
s-Energy = Pf.
∫∫M×M
|x− y|s dxdy, i.e. constant term of p(s; ε)
Residues “=” coefficients of terms of p(s; ε) with negative powers
Analytic continuation. BX(z) =
∫∫X×X
|x− y|z dxdy
s-Energy =
limz→s
(BX(z)− Res (BX , s)
z − s
)BX has a pole at s
BX(s) otherwise
Residues are residues of BX(z)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 9 / 34
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. . . . . .
Energies and residues by two regularizations
Hadamard regularization.
Laurent series p(s; ε) =
∫∫X×X\Nε(∆)
|x− y|s dxdy
s-Energy = Pf.
∫∫M×M
|x− y|s dxdy, i.e. constant term of p(s; ε)
Residues “=” coefficients of terms of p(s; ε) with negative powers
Analytic continuation. BX(z) =
∫∫X×X
|x− y|z dxdy
s-Energy =
limz→s
(BX(z)− Res (BX , s)
z − s
)BX has a pole at s
BX(s) otherwise
Residues are residues of BX(z)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 9 / 34
![Page 22: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/22.jpg)
. . . . . .
Energies and residues by two regularizations
Hadamard regularization.
Laurent series p(s; ε) =
∫∫X×X\Nε(∆)
|x− y|s dxdy
s-Energy = Pf.
∫∫M×M
|x− y|s dxdy, i.e. constant term of p(s; ε)
Residues “=” coefficients of terms of p(s; ε) with negative powers
Analytic continuation. BX(z) =
∫∫X×X
|x− y|z dxdy
s-Energy =
limz→s
(BX(z)− Res (BX , s)
z − s
)BX has a pole at s
BX(s) otherwise
Residues are residues of BX(z)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 9 / 34
![Page 23: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/23.jpg)
. . . . . .
Energies and residues by two regularizations
Hadamard regularization.
Laurent series p(s; ε) =
∫∫X×X\Nε(∆)
|x− y|s dxdy
s-Energy = Pf.
∫∫M×M
|x− y|s dxdy, i.e. constant term of p(s; ε)
Residues “=” coefficients of terms of p(s; ε) with negative powers
Analytic continuation. BX(z) =
∫∫X×X
|x− y|z dxdy
s-Energy =
limz→s
(BX(z)− Res (BX , s)
z − s
)BX has a pole at s
BX(s) otherwise
Residues are residues of BX(z)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 9 / 34
![Page 24: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/24.jpg)
. . . . . .
Energies and residues by two regularizations
Hadamard regularization.
Laurent series p(s; ε) =
∫∫X×X\Nε(∆)
|x− y|s dxdy
s-Energy = Pf.
∫∫M×M
|x− y|s dxdy, i.e. constant term of p(s; ε)
Residues “=” coefficients of terms of p(s; ε) with negative powers
Analytic continuation. BX(z) =
∫∫X×X
|x− y|z dxdy
s-Energy =
limz→s
(BX(z)− Res (BX , s)
z − s
)BX has a pole at s
BX(s) otherwise
Residues are residues of BX(z)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 9 / 34
![Page 25: Energies and residues of manifolds and configuration space ... · Purpose and outline of Lectures. The space X: a submanifold of RN; either Mm: a closed submanifold (@M = ∅and m](https://reader033.vdocument.in/reader033/viewer/2022050219/5f649d5b3d23767a8b0e2383/html5/thumbnails/25.jpg)
. . . . . .
Energies and residues by two regularizations
Hadamard regularization.
Laurent series p(s; ε) =
∫∫X×X\Nε(∆)
|x− y|s dxdy
s-Energy = Pf.
∫∫M×M
|x− y|s dxdy, i.e. constant term of p(s; ε)
Residues “=” coefficients of terms of p(s; ε) with negative powers
Analytic continuation. BX(z) =
∫∫X×X
|x− y|z dxdy
s-Energy =
limz→s
(BX(z)− Res (BX , s)
z − s
)BX has a pole at s
BX(s) otherwise
Residues are residues of BX(z)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 9 / 34
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. . . . . .
Purpose and outline of tutorials. The configuration space
Let us consider some geometric objects in Rn (usually n = 2, 3)such as polygons or (mechanical) linkages (e.g. robot arms).The configuration space (moduli space) is a space of the “shapes”
M := geometric objects/G+,
where G+ is the group of the orientation preserving isometries of Rn,G+ = SO(n)⋉Rn
We study the case when dimM is finite, especially dimM = 1, 2
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 10 / 34
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. . . . . .
Purpose and outline of tutorials. The configuration space
Let us consider some geometric objects in Rn (usually n = 2, 3)such as polygons or (mechanical) linkages (e.g. robot arms).The configuration space (moduli space) is a space of the “shapes”
M := geometric objects/G+,
where G+ is the group of the orientation preserving isometries of Rn,G+ = SO(n)⋉Rn
We study the case when dimM is finite, especially dimM = 1, 2
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 10 / 34
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. . . . . .
Example 1: Configuration space of planar pentagons
E.g.: Config. sp. of pentagons ⊂ R2 with fixed edge lengths.
(e1, . . . , e5) ∈ (R+)5 : given
P(e1, . . . , e5) = (P1, . . . , P5) ∈ (R2)5 : |PiPi−1| = ei/G+,
/G+ corresponds to fixing an edge, say P1P2
Expected dimension of P: 3 more vertices, 4 more relations (← edgelengths), hence dimP = 3× 2− 4 = 2
It is knowsn that when P is a manifold, i.e., without singularities
P ∼= S2, T 2,Σ2,Σ3,Σ4.
The genus can be computed from (e1, . . . , e5)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 11 / 34
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. . . . . .
Example 2: Config. sp. of planar “spidery linkages”
Consider mechanical linkages with arms and joints.We assume some of the joints/end points of arms are fixed.
Bi are fixed, located equally on acircle with radius RIt can move in the planeSelf-intersection is allowedAssume |BiNi| = |NiC| = 1 (∀i)
M∼=Σ17 if 1 < R < 2Σ209 if 0 < R < 1
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 12 / 34
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. . . . . .
Example 2: Config. sp. of planar “spidery linkages”
Consider mechanical linkages with arms and joints.We assume some of the joints/end points of arms are fixed.
Bi are fixed, located equally on acircle with radius RIt can move in the planeSelf-intersection is allowedAssume |BiNi| = |NiC| = 1 (∀i)
M∼=Σ17 if 1 < R < 2Σ209 if 0 < R < 1
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 12 / 34
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. . . . . .
Configuration space of 3D-linkages
Study 3D-linkages such that the configuration space is 2 dimensional.
The dimension of the config. sp. of spatial n-gons = 2⇐⇒ n = 4
Example: 3D-quadrilaterals/G+∼= S2 or torus or pinched torus
An equilateral and equiangular n-gon (α-regular stick knot) is amathematical model of cycloalkane CnH2n.
The dimension of the config. sp. = 1 (n = 6, 7) and = 2 (n = 8)
Dancing hexagons
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 13 / 34
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. . . . . .
Configuration space of 3D-linkages
Study 3D-linkages such that the configuration space is 2 dimensional.
The dimension of the config. sp. of spatial n-gons = 2⇐⇒ n = 4
Example: 3D-quadrilaterals/G+∼= S2 or torus or pinched torus
An equilateral and equiangular n-gon (α-regular stick knot) is amathematical model of cycloalkane CnH2n.
The dimension of the config. sp. = 1 (n = 6, 7) and = 2 (n = 8)
Dancing hexagons
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 13 / 34
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. . . . . .
Equilateral arccos(−1/3)-octagons
Yoshiki Kato did numerical experiments on the case whenthe bond angle = arccos(−1/3), the carbon bond angle.
.Conjecture (Kato 2019, Master Thesis in Japanese)..
......
M≈ (homeo. to) a union of two spheres with two points in common(twice pinched torus)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 14 / 34
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. . . . . .
Problems
LetM =M(e1, . . . , e8; θ1, . . . , θ8) be the config. sp. of octagonssuch that |Pi − Pi−1| = ei and ∠Pj = θj .
.Problem..
......
...1 What is the topological type ofM(1, . . . , 1; θ, . . . , θ)?
...2 When isM(e1, . . . , e8; θ1, . . . , θ8) a manifold, i.e., withoutsingularities?
...3 What are the possible genera ofM(e1, . . . , e8; θ1, . . . , θ8)?
.Problem..
......
Can Brylinski’s beta function BK(z) distinguish points inM?Cf. Can you hear the shape of a drum? (Kac)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 15 / 34
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. . . . . .
Brylinski’s beta function of a knot K
C ∋ z 7→∫∫
K×K|x− y|z dxdy ∈ C is holomorphic on ℜez > −1.
Expand the domain to C by analytic continuation. a meromorphic function with simple poles at z = −1,−3,−5, . . . .It is called Brylinski’s beta function of a knot K, denoted by BK(z)
.Theorem (Brylinski ’99)..
......BK(−2) = E(K) = Pf.
∫∫K×K
dxdy
|x− y|2
The residues are geometric quantities of a knot K;
Res(BK ,−1) = 2 Length(K)
Res(BK ,−3) = 1
4
∫K
κ2 dx
For the unit circle, B(z) = B
(z
2+
1
2,1
2
)Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 16 / 34
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. . . . . .
Brylinski’s beta function of a knot K
C ∋ z 7→∫∫
K×K|x− y|z dxdy ∈ C is holomorphic on ℜez > −1.
Expand the domain to C by analytic continuation. a meromorphic function with simple poles at z = −1,−3,−5, . . . .It is called Brylinski’s beta function of a knot K, denoted by BK(z)
.Theorem (Brylinski ’99)..
......BK(−2) = E(K) = Pf.
∫∫K×K
dxdy
|x− y|2
The residues are geometric quantities of a knot K;
Res(BK ,−1) = 2 Length(K)
Res(BK ,−3) = 1
4
∫K
κ2 dx
For the unit circle, B(z) = B
(z
2+
1
2,1
2
)Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 16 / 34
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. . . . . .
Brylinski’s beta function of a knot K
C ∋ z 7→∫∫
K×K|x− y|z dxdy ∈ C is holomorphic on ℜez > −1.
Expand the domain to C by analytic continuation. a meromorphic function with simple poles at z = −1,−3,−5, . . . .It is called Brylinski’s beta function of a knot K, denoted by BK(z)
.Theorem (Brylinski ’99)..
......BK(−2) = E(K) = Pf.
∫∫K×K
dxdy
|x− y|2
The residues are geometric quantities of a knot K;
Res(BK ,−1) = 2 Length(K)
Res(BK ,−3) = 1
4
∫K
κ2 dx
For the unit circle, B(z) = B
(z
2+
1
2,1
2
)Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 16 / 34
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. . . . . .
Brylinski’s beta function of a knot K
C ∋ z 7→∫∫
K×K|x− y|z dxdy ∈ C is holomorphic on ℜez > −1.
Expand the domain to C by analytic continuation. a meromorphic function with simple poles at z = −1,−3,−5, . . . .It is called Brylinski’s beta function of a knot K, denoted by BK(z)
.Theorem (Brylinski ’99)..
......BK(−2) = E(K) = Pf.
∫∫K×K
dxdy
|x− y|2
The residues are geometric quantities of a knot K;
Res(BK ,−1) = 2 Length(K)
Res(BK ,−3) = 1
4
∫K
κ2 dx
For the unit circle, B(z) = B
(z
2+
1
2,1
2
)Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 16 / 34
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. . . . . .
Brylinski’s beta function of a knot K
C ∋ z 7→∫∫
K×K|x− y|z dxdy ∈ C is holomorphic on ℜez > −1.
Expand the domain to C by analytic continuation. a meromorphic function with simple poles at z = −1,−3,−5, . . . .It is called Brylinski’s beta function of a knot K, denoted by BK(z)
.Theorem (Brylinski ’99)..
......BK(−2) = E(K) = Pf.
∫∫K×K
dxdy
|x− y|2
The residues are geometric quantities of a knot K;
Res(BK ,−1) = 2 Length(K)
Res(BK ,−3) = 1
4
∫K
κ2 dx
For the unit circle, B(z) = B
(z
2+
1
2,1
2
)Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 16 / 34
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. . . . . .
Brylinski’s beta function of a knot K
C ∋ z 7→∫∫
K×K|x− y|z dxdy ∈ C is holomorphic on ℜez > −1.
Expand the domain to C by analytic continuation. a meromorphic function with simple poles at z = −1,−3,−5, . . . .It is called Brylinski’s beta function of a knot K, denoted by BK(z)
.Theorem (Brylinski ’99)..
......BK(−2) = E(K) = Pf.
∫∫K×K
dxdy
|x− y|2
The residues are geometric quantities of a knot K;
Res(BK ,−1) = 2 Length(K)
Res(BK ,−3) = 1
4
∫K
κ2 dx
For the unit circle, B(z) = B
(z
2+
1
2,1
2
)Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 16 / 34
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. . . . . .
Brylinski’s beta function of a knot K
C ∋ z 7→∫∫
K×K|x− y|z dxdy ∈ C is holomorphic on ℜez > −1.
Expand the domain to C by analytic continuation. a meromorphic function with simple poles at z = −1,−3,−5, . . . .It is called Brylinski’s beta function of a knot K, denoted by BK(z)
.Theorem (Brylinski ’99)..
......BK(−2) = E(K) = Pf.
∫∫K×K
dxdy
|x− y|2
The residues are geometric quantities of a knot K;
Res(BK ,−1) = 2 Length(K)
Res(BK ,−3) = 1
4
∫K
κ2 dx
For the unit circle, B(z) = B
(z
2+
1
2,1
2
)Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 16 / 34
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. . . . . .
Brylinski beta function for polygons
BK(z) has poles at z = −1,−3,−5, . . . if K is smooth.(The domain depends on the regularity of K)
.Theorem (Brylinski ’99)..
......
If K is a polygonal knot with n vertices then BK(z) has simple poles atz = −1,−2
Res(BK ,−1) = 2 Length(K)
Res(BK ,−2) = −2k + 2
n∑j=1
π − θjsin θj
,
where θj is the angle between adjacent edges.
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 17 / 34
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. . . . . .
Brylinski beta function for polygons
BK(z) has poles at z = −1,−3,−5, . . . if K is smooth.(The domain depends on the regularity of K)
.Theorem (Brylinski ’99)..
......
If K is a polygonal knot with n vertices then BK(z) has simple poles atz = −1,−2
Res(BK ,−1) = 2 Length(K)
Res(BK ,−2) = −2k + 2
n∑j=1
π − θjsin θj
,
where θj is the angle between adjacent edges.
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 17 / 34
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. . . . . .
Brylinski beta function for polygons
BK(z) has poles at z = −1,−3,−5, . . . if K is smooth.(The domain depends on the regularity of K)
.Theorem (Brylinski ’99)..
......
If K is a polygonal knot with n vertices then BK(z) has simple poles atz = −1,−2
Res(BK ,−1) = 2 Length(K)
Res(BK ,−2) = −2k + 2
n∑j=1
π − θjsin θj
,
where θj is the angle between adjacent edges.
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 17 / 34
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. . . . . .
Brylinski beta function for polygons
BK(z) has poles at z = −1,−3,−5, . . . if K is smooth.(The domain depends on the regularity of K)
.Theorem (Brylinski ’99)..
......
If K is a polygonal knot with n vertices then BK(z) has simple poles atz = −1,−2
Res(BK ,−1) = 2 Length(K)
Res(BK ,−2) = −2k + 2
n∑j=1
π − θjsin θj
,
where θj is the angle between adjacent edges.
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 17 / 34
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. . . . . .
Motivation for the energy for knots
.Problem (Fukuhara, Sakuma)..
......
Find a functional (which we call an energy) on knots so that for everyknot type we can get an “optimal configuration” as an energy minimizer.
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 18 / 34
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. . . . . .
Our strategy
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 19 / 34
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. . . . . .
Our strategy
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 20 / 34
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. . . . . .
Our strategy
Each “cell ” correspondsto a knot type.
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 21 / 34
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. . . . . .
Our strategy
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 22 / 34
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. . . . . .
Our strategy
Deform it along thegradient flow of the“energy” e.
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 23 / 34
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. . . . . .
Our strategy
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 24 / 34
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. . . . . .
Our strategy
Crossing changes duringthe deformation processshould be avoided!
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 25 / 34
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. . . . . .
Our strategy
We require that ourfunctional +∞as K degenerates to havedouble points.
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 26 / 34
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. . . . . .
Definition of an energy of knots
.Definition..
......
A functional e : knots → R is called self-repulsiveif it blows up as a knot degenerates to have double points.
=
.Definition..
......
A functional e : knots → R is called an energy if it is
(i) self-repulsive,(ii) bounded below,(iii) continuous in some sense, say w.r.t. C2-top.
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 27 / 34
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. . . . . .
Definition of an energy of knots
.Definition..
......
A functional e : knots → R is called self-repulsiveif it blows up as a knot degenerates to have double points.
=
.Definition..
......
A functional e : knots → R is called an energy if it is
(i) self-repulsive,(ii) bounded below,(iii) continuous in some sense, say w.r.t. C2-top.
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 27 / 34
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. . . . . .
How to get an energy for knots
Candidate: an electrostatic energy of a charged knot
+ +++
+
+
+
+
+ ++
+
+
+
+
++
+ +
++
++
++
+
+
+
+
+
+
++
+ +
++
++
+
+
+
+
+
|Coulomb’s force| ∝ 1
r2, potential energy =
∫∫K×K
dx dy
|x− y|Apply regularization (HR or AC)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 28 / 34
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. . . . . .
How to get an energy for knots
Candidate: an electrostatic energy of a charged knot
+ +++
+
+
+
+
+ ++
+
+
+
+
++
+ +
++
++
++
+
+
+
+
+
+
++
+ +
++
++
+
+
+
+
+
|Coulomb’s force| ∝ 1
r2, potential energy =
∫∫K×K
dx dy
|x− y|Apply regularization (HR or AC)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 28 / 34
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. . . . . .
How to get an energy for knots
Candidate: an electrostatic energy of a charged knot
+ +++
+
+
+
+
+ ++
+
+
+
+
++
+ +
++
++
++
+
+
+
+
+
+
++
+ +
++
++
+
+
+
+
+
|Coulomb’s force| ∝ 1
r2, potential energy =
∫∫K×K
dx dy
|x− y|=∞ (∀K)
Apply regularization (HR or AC)
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 28 / 34
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. . . . . .
How to get an energy for knots
Candidate: an electrostatic energy of a charged knot
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|Coulomb’s force| ∝ 1
r2, potential energy =
∫∫K×K
dx dy
|x− y|=∞ (∀K)
Apply regularization (HR or AC)
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. . . . . .
Energy of knots
An electrostatic energy of a charged knot
∫∫K×K
dx dy
|x− y|
Hadamard regularization Pf.
∫∫K×K
dx dy
|x− y|is not self-repulsive
Increase the power.Self-repulsive if the power ≥ 2.
.Definition..
......E(K) := Pf.
∫∫K×K
dx dy
|x− y|2
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. . . . . .
Energy of knots
An electrostatic energy of a charged knot
∫∫K×K
dx dy
|x− y|
Hadamard regularization Pf.
∫∫K×K
dx dy
|x− y|is not self-repulsive
Increase the power.Self-repulsive if the power ≥ 2.
.Definition..
......E(K) := Pf.
∫∫K×K
dx dy
|x− y|2
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 29 / 34
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. . . . . .
Energy of knots
An electrostatic energy of a charged knot
∫∫K×K
dx dy
|x− y|
Hadamard regularization Pf.
∫∫K×K
dx dy
|x− y|is not self-repulsive
Increase the power.Self-repulsive if the power ≥ 2.
.Definition..
......E(K) := Pf.
∫∫K×K
dx dy
|x− y|2
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. . . . . .
Mobius transformations ∼ inversion in a circle
Inversion in the unit circle of C ∪ ∞ is given by C ∋ z 7→ 1
z
It is angle-preserving (conformal, i.e. “microscopically homothetic”),and it maps circles (including lines) to circles (including lines).
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. . . . . .
Mobius transformation
Inversion in a sphere Σ with center C and radius r
P 7→
∞ (P = C)C (P =∞)P ′ (P = C,P ), P ′ ∈ half line CP, |CP ||CP ′| = r2
r
C
P
P
‘
A Mobius transformation of R3 ∪ ∞ is a transformation of R3 ∪ ∞that can be obtained as a composition of inversions in spheres(including reflections in planes).
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. . . . . .
Mobius invariance of the energy of knots
Recall E(K) = Pf.
∫∫K×K
dx dy
|x− y|2.Theorem (Freedman-He-Wang ’94)..
......
The energy E is invariant under Mobius transformations;E(T (K)) = E(K) for any Mobius transformation T and for any knot K
.Corollary........For any prime knot type there is an E-minimizer.
prime = not composite
.Theorem (Freedman-He-Wang ’94)........The round circle gives theminimum value of E.
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. . . . . .
Energy minimizers by Rob Kusner and John M. Sullivan
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. . . . . .
Related topics
Regularity of E-minimizers. (Zheng-Xu He, Simon Blatt, PhilippReiter, Armin Schikorra, Aya Ishizeki, Takeyuki Nagasawa, AlexandraGilsbach, Heiko von der Mosel, and Nicole Vorderobermeier)
Other energies of knots
Energy for higher dimensional manifolds (surfaces etc)
Functionals that measure geometric complexity of mfds.
Numerical experiments
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 34 / 34
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. . . . . .
Related topics
Regularity of E-minimizers. (Zheng-Xu He, Simon Blatt, PhilippReiter, Armin Schikorra, Aya Ishizeki, Takeyuki Nagasawa, AlexandraGilsbach, Heiko von der Mosel, and Nicole Vorderobermeier)
Other energies of knots
Energy for higher dimensional manifolds (surfaces etc)
Functionals that measure geometric complexity of mfds.
Numerical experiments
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 34 / 34
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. . . . . .
Related topics
Regularity of E-minimizers. (Zheng-Xu He, Simon Blatt, PhilippReiter, Armin Schikorra, Aya Ishizeki, Takeyuki Nagasawa, AlexandraGilsbach, Heiko von der Mosel, and Nicole Vorderobermeier)
Other energies of knots
Energy for higher dimensional manifolds (surfaces etc)
Functionals that measure geometric complexity of mfds.
Numerical experiments
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 34 / 34
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. . . . . .
Related topics
Regularity of E-minimizers. (Zheng-Xu He, Simon Blatt, PhilippReiter, Armin Schikorra, Aya Ishizeki, Takeyuki Nagasawa, AlexandraGilsbach, Heiko von der Mosel, and Nicole Vorderobermeier)
Other energies of knots
Energy for higher dimensional manifolds (surfaces etc)
Functionals that measure geometric complexity of mfds.
Numerical experiments
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 34 / 34
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. . . . . .
Related topics
Regularity of E-minimizers. (Zheng-Xu He, Simon Blatt, PhilippReiter, Armin Schikorra, Aya Ishizeki, Takeyuki Nagasawa, AlexandraGilsbach, Heiko von der Mosel, and Nicole Vorderobermeier)
Other energies of knots
Energy for higher dimensional manifolds (surfaces etc)
Functionals that measure geometric complexity of mfds.
Numerical experiments
Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 34 / 34