Presentation Jun O'Hara orea

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. . . . . .

.

.. . .

.

. Energy of knots and related topics

Jun O’Hara ^（Tokyo Metropolitan University^）

28/07/2010

The 2nd TAPU-KOOK Joint Seminar on Knots and Related Topics

& The 4th Graduate Student Workshop on Mathematics Kyungpook National University

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(Kusner and J.Sullivan)

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. . . . . .

.. Table of contents

Energy of knots (first 10 years)

Motivation ∼ optimal configuration for each knot type. Renormalization

M¨obius invariance and existence of energy minimizers Various energies

Renormalization of Riesz-type potential and generalization of barycenter (coming 10 years ?)

Renormalized r⁻⁴ potential energy of surfaces and the average of the squared linking numbers with random circles (with Gil Solanes) (coming 10 years ?)

(If time permits) M¨obius geometry and the infinitesimal cross ratio (with R´emi Langevin). (last 10 years)

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. . . . . .

.. Table of contents

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. . . . . .

.. Energy of knots

Motivation (Fukuhara, Sakuma): Produce “optimal configurations” for every knot type as energy minimizers. A functional e : {knots} → R is called an energy if it blows up as a knot degenerates to a singular knot with double points.

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Motivation (Fukuhara, Sakuma): Produce “optimal configurations” for every knot type as energy minimizers. A functional e : {knots} → R is called an energy if it blows up as a knot degenerates to a singular knot with double points.

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. . . . . .

.. Our strategy

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. . . . . .

.. Our strategy

Each “cell” corresponds to a knot type.

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. . . . . .

.. Our strategy

Deform it along the gradient flow of the

“energy” e.

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. . . . . .

.. Our strategy

Crossing changes during the deformation process should be avoided!

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e : {knots} → R is an energy of knots

⇐⇒ e(K) blows up as K degenerates to a singular knot with double points.

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. . . . . .

.. How to define energy of knots

The “voltage” at point x and “potential energy” are given by

“V (K; x)” =

∫

K

dy

|x − y|

“E(K)” =

∫

K

V (x) dx =

∫∫

K_×K

dxdy

|x − y|^.

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The “voltage” at point x and “potential energy” are given by

“V (K; x)” =

∫

K

dy

|x − y| ^{= ∞}

“E(K)” =

∫

K

V (x) dx =

∫∫

K_×K

dxdy

|x − y| ^{= ∞.} Thus we need renormalization

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. . . . . .

.. Renormalization

Suppose

∫

Ω

ω blows up on X ⊂ Ω.

Remove the ε-tub. nbd. Nε(X) and put p(ε) :=

∫

Ω\Nε(X)

ω. Then lim

ε_→+0^{p(ε) = ∞.}

Expand p(ε) in a series of ¹

ε^{: p(ε) = a}⁰⁺ a₁

ε ⁺ a₂ ε² ^{+ · · ·} The constant a₀ is what we get by the renormalization. Sometimes we need a log¹_ε term.

Sometimes the series may be a₀+ ¹ ε^b

( a^′₀+^a

′1

ε ⁺ a^′₂ ε² ^{+ · · ·}

)

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Suppose

∫

Ω

∫

Ω\Nε(X)

ω. Then lim

ε_→+0^{p(ε) = ∞.}

ε^{: p(ε) = a}⁰⁺ a₁

( a^′₀+^a

′1

ε ⁺ a^′₂ ε² ^{+ · · ·}

)

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. . . . . .

.. Renormalization

Suppose

∫

Ω

∫

Ω\Nε(X)

ω. Then lim

ε_→+0^{p(ε) = ∞.}

ε^{: p(ε) = a}⁰⁺ a₁

( a^′₀+^a

′1

ε ⁺ a^′₂ ε² ^{+ · · ·}

)

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Suppose

∫

Ω

∫

Ω\Nε(X)

ω. Then lim

ε_→+0^{p(ε) = ∞.}

ε^{: p(ε) = a}⁰⁺ a₁

( a^′₀+^a

′1

ε ⁺ a^′₂ ε² ^{+ · · ·}

)

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. . . . . .

.. Renormalization

Suppose

∫

Ω

∫

Ω\Nε(X)

ω. Then lim

ε_→+0^{p(ε) = ∞.}

ε^{: p(ε) = a}⁰⁺ a₁

( a^′₀+^a

′1

ε ⁺ a^′₂ ε² ^{+ · · ·}

)

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V (K; x) = lim

ε_→+0

(∫

K_\(ε-nbd. of x)

dy

|x − y|^{− 2 log} 1 ε

)

E(K) =

∫

K

V (K; x) dx

= lim

ε_→+0

(∫∫

K_×K\Nε(∆)

dxdy

|x − y|− 2L(K) log¹ ε

)

charged

But this E is not an energy, i.e it is finite for singular knots with double points.

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. . . . . .

.. Renormalized energy

The renormalization can be done as V (K; x) = lim

ε_→+0

(∫

K_\(ε-nbd. of x)

dy

|x − y|^{− 2 log} 1 ε

)

E(K) =

∫

K

V (K; x) dx

= lim

ε_→+0

(∫∫

K_×K\Nε(∆)

dxdy

|x − y|− 2L(K) log¹ ε

)

charged

But this E is not an energy, i.e it is finite for singular knots with double points.

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Let α > 1. Put V^(α)(K; x) = lim

ε_→+0

(∫

K_\N_ε_(x)

dy

|x − y|^α ⁻

(a function F of ε and x

))

E^(α)(K) =

∫

K

V^(α)(K; x) dx.

When α < 3 we can take the counter term F (ε, x) by F (ε, x) = ²

(α − 1)ε^α⁻¹, i.e. independent of x.

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. . . . . .

.. ^r

^−α

-modified energy (1)

Let α > 1. Put V^(α)(K; x) = lim

ε_→+0

(∫

K_\N_ε_(x)

dy

|x − y|^α ⁻

(a function F of ε and x

))

E^(α)(K) =

∫

K

V^(α)(K; x) dx.

When α < 3 we can take the counter term F (ε, x) by F (ε, x) = ²

(α − 1)ε^α⁻¹, i.e. independent of x.

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When 1 < α < 3 the renormalization can be done as V^(α)(K; x) = lim

ε_→+0

(∫

K_\N_ε_(x)

dy

|x − y|^α ⁻

2 (α − 1)ε^α⁻¹

)

E^(α)(K) =

∫

K

V^(α)(K; x) dx

= lim

ε_→+0

(∫

K_×K\Nε(∆)

dxdy

|x − y|^α ⁻

2L(K) (α − 1)ε^α⁻¹

)

When α ≥ 3, F (ε, x) depends on x.

E^(α) thus defined (1 < α < 3) is an energy if and only if α ≥ 2.

Let us study E⁽²⁾.

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. . . . . .

.. ^r

^−α

-modified energy (2)

ε_→+0

(∫

K_\N_ε_(x)

dy

|x − y|^α ⁻

2 (α − 1)ε^α⁻¹

)

E^(α)(K) =

∫

K

V^(α)(K; x) dx

= lim

ε_→+0

(∫

K_×K\Nε(∆)

dxdy

|x − y|^α ⁻

2L(K) (α − 1)ε^α⁻¹

)

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ε_→+0

(∫

K_\N_ε_(x)

dy

|x − y|^α ⁻

2 (α − 1)ε^α⁻¹

)

E^(α)(K) =

∫

K

V^(α)(K; x) dx

= lim

ε_→+0

(∫

K_×K\Nε(∆)

dxdy

|x − y|^α ⁻

2L(K) (α − 1)ε^α⁻¹

)

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. . . . . .

.. Properties of E

⁽²⁾

.Theorem (Freedman-He-Wang) .

.

... .

.

E_◦⁽²⁾ is invariant under M¨obius transformations, i.e. if T is a M¨obius transformation of R³∪ {∞} then E⁽²⁾(T (K)) = E⁽²⁾(K) ∀K.

The answer to our motivational problem depends on a topological condition.

.

... .

. .There is an E⁽²⁾-minimizer for each prime knot type.

.Conjecture (Kusner-J.Sullivan) .

.

... .

. .There are no E⁽²⁾-minimizers for composite knot types.

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Theorem (Freedman-He-Wang) .

.

... .

.

... .

.

... .

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. . . . . .

.. Properties of E

⁽²⁾

.

... .

.

... .

.

... .

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Theorem (Freedman-He-Wang) .

.

... .

.

... .

.

... .

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. . . . . .

.. How composite knots behave ∼ pull-tight

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. . . . . .

.. Prime knots Case

Why do prime knots have E◦⁽²⁾-minimizers? How can they avoid pull-tight?)

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He shows that E⁽²⁾-minimizers are smooth.

There are uncountably many E⁽²⁾-minimizers for each non-trivial prime knot type.

{ K : an E_◦⁽²⁾-minimizer of a knot type [K] T : a M¨obius transformation

⇒

{ T (K) or its mirror image T (K)^∗ ∈ [K], and E⁽²⁾(T (K)) = E⁽²⁾(T (K)^∗) = E⁽²⁾(K).

⇒ T (K) or T (K)^∗ is an E⁽²⁾-minimizer of [K]. .Open problem

. .

... .

. .♯ ({Minimizers of [K]}/M¨obius group) =?

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. . . . . .

.. ^{Remarks I}

⇒

. .

... .

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⇒

. .

... .

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. . . . . .

.. ^{Remarks I}

⇒

. .

... .

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.Open problem .

.

... .

.

.∃ E⁽²⁾_◦ -ciritical unknots besides round circles? If NO =⇒ Hatcher’s results:

{unknots in S³^}−−−−−−−−−−→^≃ deform. retract

{great circles in S³^}

E⁽²⁾ can untie the following “unknots”: Ochiai’s unknot (Huang, Kauffman, and Grzeszczuk ’97) and “Freedman’s unknot” (Kusner and Sullivan ’94).

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. . . . . .

.. ^{Remarks II}

.Open problem .

.

... .

.

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.Open problem .

.

... .

.

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. . . . . .

.. ^Unknots

Figure: Ochiai’s unknot

Figure: Freedman’s unknot

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(Kusner and J.Sullivan)

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. . . . . .

.. How to produce energy minimizers

Find an energy for which ∀ knot type has an energy minimizer. Two solutions.

E⁽²⁾(K) = −4 +

∫∫

K_×K

( 1

|x − y|² ⁻ 1 dK(x, y)²

) dxdy

Change the power 2 bigger

Change the metric of the ambient space In each case, no M¨obius invariance any more.

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E⁽²⁾(K) = −4 +

∫∫

K_×K

( 1

|x − y|² ⁻ 1 dK(x, y)²

) dxdy

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. . . . . .

.. How to produce energy minimizers

E⁽²⁾(K) = −4 +

∫∫

K_×K

( 1

|x − y|² ⁻ 1 dK(x, y)²

) dxdy

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E⁽²⁾(K) = −4 +

∫∫

K_×K

( 1

|x − y|² ⁻ 1 dK(x, y)²

) dxdy

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. . . . . .

.. How to produce energy minimizers

E⁽²⁾(K) = −4 +

∫∫

K_×K

( 1

|x − y|² ⁻ 1 dK(x, y)²

) dxdy

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Suppose K has length 1. When 2 ≤ α < 3

E^(α)(K) = lim

ε_→+0

(∫

K_×K\N_ε_(∆)

dxdy

|x − y|^α ⁻

2 (α − 1)ε^α⁻¹

)

.Theorem .

.

... .

. .There is anE^(α)-minimizer for any knot type if(3 >)α > 2.

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. . . . . .

.. Energy with bigger index

Suppose K has length 1. When 2 ≤ α < 3

E^(α)(K) = lim

ε_→+0

(∫

K_×K\N_ε_(∆)

dxdy

|x − y|^α ⁻

2 (α − 1)ε^α⁻¹

)

.Theorem .

.

... .

. .There is anE^(α)-minimizer for any knot type if(3 >)α > 2.

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Ω : a compact domain in R^m, α ∈ R.

R^m _{∋ x 7→ V}^(α)

Ω ^{(x) :=}

∫

Ω

|x − y|^α^−mdµ(y) It is well-defined if α > 0.

It is called the Riesz potential if 0 < α < m. If α ≤ 0 and x ∈ Ω the integral blows up near x. The same method of renormalization applies

⇝ V_Ω^(α)^{: R}^m\ ∂Ω → R (∀α)

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. . . . . .

.. Renormalization of Riesz-type potential

R^m _{∋ x 7→ V}^(α)

Ω ^{(x) :=}

∫

Ω

⇝ V_Ω^(α)^{: R}^m\ ∂Ω → R (∀α)

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R^m _{∋ x 7→ V}^(α)

Ω ^{(x) :=}

∫

Ω

⇝ V_Ω^(α)^{: R}^m\ ∂Ω → R (∀α)

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. . . . . .

.. Renormalization of Riesz-type potential

R^m _{∋ x 7→ V}^(α)

Ω ^{(x) :=}

∫

Ω

⇝ V_Ω^(α)^{: R}^m\ ∂Ω → R (∀α)

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K. Shibata introduced an “illuminating center” of a triangle. .Problem

. .

... .

. .Where should we place a street lamp in a triangular park?

The illuminating center of a planar domain Ω is the point that maximizes V_Ω⁽⁰⁾ in _Ω.^◦

V_Ω⁽⁰⁾(x) = lim

ε_→+0

(∫

Ω\Bε(x)

dµ(y)

|x − y|² ^{− 2π log} 1 ε

)

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. . . . . .

.. Illuminating center

. .

... .

ε_→+0

(∫

Ω\Bε(x)

dµ(y)

|x − y|² ^{− 2π log} 1 ε

)

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. .

... .

ε_→+0

(∫

Ω\Bε(x)

dµ(y)

|x − y|² ^{− 2π log} 1 ε

)

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. . . . . .

.. Generalization of barycenter

The barycenter (center of mass) ¯x of a domain Ω ⊂ R^m is given by

¯ x =

∫

Ω

y dµ(y)

∫

Ω

1 dµ(y) .

It is characterized by

∫

Ω

(¯x − y) dµ(y) = 0 ⇐⇒

∫

Ω

(¯xi− yi) dµ(y) = 0. The barycenter is the unique point that minimizes

V_Ω^(m+2)(x) =

∫

Ω

|x − y|²dµ(y) =

∫

Ω

∑m i₌₁

(xi_{− y}i)²dµ(y)

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¯ x =

∫

Ω

y dµ(y)

∫

Ω

1 dµ(y) .

∫

Ω

(¯x − y) dµ(y) = 0 ⇐⇒

∫

Ω

V_Ω^(m+2)(x) =

∫

Ω

|x − y|²dµ(y) =

∫

Ω

∑m i₌₁

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. . . . . .

.. Generalization of barycenter

¯ x =

∫

Ω

y dµ(y)

∫

Ω

1 dµ(y) .

∫

Ω

(¯x − y) dµ(y) = 0 ⇐⇒

∫

Ω

V_Ω^(m+2)(x) =

∫

Ω

|x − y|²dµ(y) =

∫

Ω

∑m i₌₁

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. .

... . ^.

An r^α^−m-centerof Ω ⊂ R^m is a point that attains the exremal value of V_Ω^(α).

To be precise, according to α,

the minimum value of V_Ω^(α) when α > m, the maximum value of V_Ω^(α) when 0 < α < m, the maximum value of V_Ω^(α)^◦

Ωwhen α ≤ 0. The illuminating center is an r⁻²-center.

The barycenter (center of mass) is an r²-center. .Theorem

. .

... .

.

Any compact domain with some natural technical conditions has a r^α^−m-center for any α ̸= m.

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. . . . . .

.. Generalized centers

.Definition .

.

... .

.

. .

... .

.

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. .

... . ^.

. .

... .

.

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. . . . . .

.. Generalized centers

.Definition .

.

... .

.

. .

... .

.

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. .

... . ^.

. .

... .

.

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. . . . . .

.. Generalized centers

.Definition .

.

... .

.

. .

... .

.

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. .

... . ^.

. .

... .

.

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. . . . . .

.. ^r

^α−m

-center of a convex set

r^a-center is not necessarily unique. Example: Ω = [−R, −1] ∪ [1, R] ⊂ R.

An r^a-center (a > 1) is the origin. {r¹-center} = [−1, 1].

∃ two r^a-centers (a < 1, a ̸= 0) in (−R, −1) and (1, R). If α ≥ m + 1 then any compact domain has a unique r^α^−m-center.

.Theorem .

.

... .

.

If a compact domainΩ ⊂ R^m is convex, then the r^α^−m-center is unique ifα ≤ 1.

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.Theorem .

.

... .

.

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. . . . . .

.. ^r

^α−m

-center of a convex set

.Theorem .

.

... .

.

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.Theorem .

.

... .

.

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. . . . . .

.. ^r

^α−m

-center of a convex set

.Theorem .

.

... .

.

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.Theorem .

.

... .

.

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. . . . . .

.. ^r

^α−m

-center of a convex set

.Theorem .

.

... .

.

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(with Gil Solanes) Suppose m = 2 and α = −2. Ω₁, Ω₂ ⊂ R² (Ω₁∩ Ω2^{= ∅).}

E(Ω1, Ω2) :=

∫

Ω1

V_Ω⁽⁻²⁾

2 (x) dµ(x) =

∫

Ω1×Ω2

dµ(x) dµ(y)

|x − y|⁴ Recall ωcr is the infinitesimal cross ratio of C ∼= R².

E(Ω₁, Ω₂) =¹ 2

∫

Ω1×Ω2

ℜe ωcr∧ ℜe ωcr

=¹ 2

∫

Ω₁×Ω₂

ℑm ωcr∧ ℑm ωcr

Put Ω1 = Ω2 =: Ω. Then we need renormalization.

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. . . . . .

.. ^r

⁻⁴

-potential energy and infinitesimal cross ratio

E(Ω1, Ω2) :=

∫

Ω1

V_Ω⁽⁻²⁾

2 (x) dµ(x) =

∫

Ω1×Ω2

dµ(x) dµ(y)

E(Ω₁, Ω₂) =¹ 2

∫

Ω1×Ω2

=¹ 2

∫

Ω₁×Ω₂

(80)

E(Ω1, Ω2) :=

∫

Ω1

V_Ω⁽⁻²⁾

2 (x) dµ(x) =

∫

Ω1×Ω2

dµ(x) dµ(y)

E(Ω₁, Ω₂) =¹ 2

∫

Ω1×Ω2

=¹ 2

∫

Ω₁×Ω₂

(81)

. . . . . .

.. ^r

⁻⁴

-potential energy and infinitesimal cross ratio

E(Ω1, Ω2) :=

∫

Ω1

V_Ω⁽⁻²⁾

2 (x) dµ(x) =

∫

Ω1×Ω2

dµ(x) dµ(y)

E(Ω₁, Ω₂) =¹ 2

∫

Ω1×Ω2

=¹ 2

∫

Ω₁×Ω₂

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When x ∈Ω⊂ R^◦ ²

V_Ω⁽⁻²⁾(x) = lim

ε_→+0

(∫

Ω\Bε(x)

dµ(y)

|x − y|⁴ ⁻ π ε²

)

V_Ω⁽⁻²⁾(x) → −∞ as x → ∂Ω.

Another renormalization is needed for

∫

Ω

V_Ω⁽⁻²⁾(x) dµ(x) = −∞ to define E(Ω);

E(Ω) = lim

δ_→+0

(∫

Ω\Nδ(∂Ω)

V_Ω⁽⁻²⁾(x) dµ(x) + ^π

4δ^L(∂Ω) )

.

(83)

. . . . . .

.. Renormalized r

⁻⁴

-potential energy

V_Ω⁽⁻²⁾(x) = lim

ε_→+0

(∫

Ω\Bε(x)

dµ(y)

|x − y|⁴ ⁻ π ε²

)

V_Ω⁽⁻²⁾(x) → −∞ as x → ∂Ω.

∫

Ω

E(Ω) = lim

δ_→+0

(∫

Ω\Nδ(∂Ω)

V_Ω⁽⁻²⁾(x) dµ(x) + ^π

4δ^L(∂Ω) )

.

(84)

V_Ω⁽⁻²⁾(x) = lim

ε_→+0

(∫

Ω\Bε(x)

dµ(y)

|x − y|⁴ ⁻ π ε²

)

V_Ω⁽⁻²⁾(x) → −∞ as x → ∂Ω.

∫

Ω

E(Ω) = lim

δ_→+0

(∫

Ω\Nδ(∂Ω)

V_Ω⁽⁻²⁾(x) dµ(x) + ^π

4δ^L(∂Ω) )

.

(85)

. . . . . .

.. Average squared linking number with randomm circles

Let K be a knot in R³ and Ω be a Seifert surface of K. Generalize the renormalized r⁻⁴-potential energy to this Seifert surface Ω to get E(K).

.Theorem (with Gil Solanes) .

.

... .

.

E(K) can be expressed by the intgral on K × K.

It is the renormalization of the average of the square of the linking number of K and random circles with respect to M¨obius group.

(86)

.

... .

.

(87)

. . . . . .

.. Average squared linking number with randomm circles

.

... .

.

(88)

(with R´emi Langevin) The cross ratio of 4 complex numbers w, w + dw, z, and z + dz is given by

(w + dw) − w (w + dw) − (z + dz) ^:

z − w z − (z + dz) ⁼

dwdz (w − z)² ^. It is a 2-form on C × C \ ∆. Denote it by ωcr.

Put ¯^C= C ∪ {∞} ∼= S². Then ωcr can be considered as a 2-form on ¯^C× ¯^C\ ∆ ∼= S²× S²\ ∆.

Note that ωcr is invariant under diagonal action of M¨obius transformations.

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. . . . . .

.. Infinitesimal cross ratio of C ∪ {∞} ∼ ₌ ^S

²

(w + dw) − w (w + dw) − (z + dz) ^:

z − w z − (z + dz) ⁼

(90)

(w + dw) − w (w + dw) − (z + dz) ^:

z − w z − (z + dz) ⁼

(91)

. . . . . .

.. Infinitesimal cross ratio of C ∪ {∞} ∼ ₌ ^S

²

(w + dw) − w (w + dw) − (z + dz) ^:

z − w z − (z + dz) ⁼

(92)

S²× S²\ ∆ can be identified with T^∗S²

Each cotangent bundle T^∗M has the canonical symplectic form ωM.

.Theorem (Folklore) .

.

... .

.

The real part of infinitesimal cross ratio coincides with the canonical symplectic form of the cotangent bundle T^∗S².

The real part can be generalized to the canonical symplectic form of

T^∗Sⁿ^∼= Sⁿ× Sⁿ\ ∆ ∼= {S⁰⊂ Sⁿ}

∼= SO(n + 1, 1)/SO(n) × SO(1, 1).

Presentation Jun O'Hara orea

.

.. . .

.

. Energy of knots and related topics

.. Table of contents

.. Table of contents

.. Energy of knots

.. Our strategy

.. Our strategy

.. Our strategy

.. Our strategy

.. How to define energy of knots

.. Renormalization

.. Renormalization

.. Renormalization

.. Renormalized energy

.. r

-modified energy (1)

.. r

-modified energy (2)

.. Properties of E

.. Properties of E

.. How composite knots behave ∼ pull-tight

.. Prime knots Case

.. Remarks I

.. Remarks I

.. Remarks II

.. Unknots

.. How to produce energy minimizers

.. How to produce energy minimizers

.. How to produce energy minimizers

.. Energy with bigger index

.. Renormalization of Riesz-type potential

.. Renormalization of Riesz-type potential

.. Illuminating center

.. Generalization of barycenter

.. Generalization of barycenter

.. Generalized centers

.. Generalized centers

.. Generalized centers

.. r

-center of a convex set

.. r

-center of a convex set

.. r

-center of a convex set

.. r

-center of a convex set

.. r

-potential energy and infinitesimal cross ratio

.. r

-potential energy and infinitesimal cross ratio

.. Renormalized r

-potential energy

.. Average squared linking number with randomm circles

.. Average squared linking number with randomm circles

.. Infinitesimal cross ratio of C ∪ {∞} ∼ = S

.. Infinitesimal cross ratio of C ∪ {∞} ∼ = S

.. ^r

.. ^r

.. ^{Remarks I}

.. ^{Remarks I}

.. ^{Remarks II}

.. ^Unknots

.. ^r

.. ^r

.. ^r

.. ^r

.. ^r

.. ^r

.. Infinitesimal cross ratio of C ∪ {∞} ∼ ₌ ^S

.. Infinitesimal cross ratio of C ∪ {∞} ∼ ₌ ^S