. . . . . .

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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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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).

Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 30 / 34

. . . . . .

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).

Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 31 / 34

. . . . . .

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.

Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 32 / 34

. . . . . .

Energy minimizers by Rob Kusner and John M. Sullivan

Jun O’Hara (Chiba University) Energies and residues of manifolds and configuration space of polygons Plan of Lectures and TutorialsJune 2019 33 / 34

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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