Magnetic knots and groundstate energy spectrum:

Lecture 3

Magnetic knots and groundstate energy spectrum:

- Magnetic relaxation

- Topology bounds the energy

- Inflexional instability of magnetic knots

- Constrained minimization of magnetic energy

- Groundstate energy spectra of magnetic knots and links - Bending energy spectra: Magnetic vs. elastic systems

Selected references

Maggioni, F & Ricca, RL 2009 On the groundstate energy of tight knots.

Proc R Soc A 465, 2761.

Ricca, RL 2008 Topology bounds energy of knots and links. Proc R Soc A 464, 293.

Ricca, RL & Maggioni, F 2014 On the groundstate energy spectrum of

magnetic knots and links. J Phys A: Math & Theor 47, 205501.

Magnetic relaxation

Magnetic relaxation (Moffatt 1985): consider in a viscous and perfectly conducting fluid;

Near equilibrium we have:

Thus:

with magnetic energy monotonically decreasing as . t → ∞

dM

dt = B ⋅ ∂ ^B

∂ ^t

V B

( ) ∫ ^d

^{X =} B ⋅ ∇ × u × B ( )

V B

( ) ∫ ^d

^X

= − u ⋅ ∇ × B ( ) ^{× B}

V B

( ) ∫ ^d

^{X = − u ⋅ J × B [ ]}

V B

( ) ∫ ^d

^X

Du

Dt ∝ u

hence

dM

dt ∝ − u

V

( ) ∫

^d

X ⇒ M ( L

) ^↓

Du

Dt = − ∇p + J × B ⇒ u ∝ −∇p + J × B ϕ :L

→ L

.

.

Magnetic relaxation to groundstate

Lk

= 0 ∀i

N components same flux

Magnetic helicity:

H = A ⋅ B

V (B)

∫ ^d

^{x = Φ}

^Lk

i≠ j

∑

Φ Φ

Φ

Φ

Φ

ϕ ϕ

knot tightening link tightening

Magnetic relaxation (Moffatt 1985) under -preserving flow: { V, Φ

}

Theorem 1 (Freedman 1988; Moffatt 1990).

i) ∃

l.b.: lim ; ii) ,

t→∞

M t ( ) ^{= M}

^M

= m Φ

V

where is a topological invariant of . m L

.

Φ

= Φ .

Φ

ϕ :L

→ L

Let be a “zero-framed” magnetic link:

Topology bounds the energy

Theorem 2 (Arnold 1974; Freedman & He 1991).

i) ; ii) , where depends on the geometry of .

M t ( ) ^{≥ q H}

q > 0 supp B ( )

M t ( ) ^{≥ 2}

π

1/ 3

Φ

C V

L ⁿ

Theorem (Ricca 2008). Let be a zero-framed link. Then, we have:

I) : ,

II) : . M

= 2 π

1/3

c

Φ

V

M t ( ) ^{≥ 2}

π

1/ 3

1

V

H

m = 2 π

1/3

c

q = 2

π

1/3

1 V

Proof:

From and we have Φ

= Φ ^Lk

= 0 ∀i ∈ 1,..., n ( )

from C = ε

and , we have

r

∑

i≠ j

∑ ^∑

^Lk

^{= 1} ₂ ^∑

^∑

^ε

H = Φ

Lk

;

i = j

∑ ^{+ Lk}

i≠ j

∑ ^{= Φ}

^Lk

i≠ j

= 0 ∑

. Therefore

. Now, from Theorem 2(ii), we have:

or

;

H = Φ

Lk

i≠ j

∑ ^{≤ Φ}

^{C C ≥} _Φ ^H

M ≥ 2 π

1/ 3

Φ

C

V

≥ 2 π

1/ 3

Φ

c

V

also, by using above, we have:

Hence, from Theorem 2(i) follows (I). Moreover, at , from Theorem 1(ii) we have (II).

.

M

2 π

1/3

Φ

C

V

≥ 2 π

1/3

Φ

V

H

Φ

= 2 π

1/3

H V

(∗)

(∗)

In general, we have:

Question: do knot types of same c

relax to same groundstate energy?

M

∝ c

C ≥ ε

r

∑

i≠ j

∑ ^{= Lk}

i≠ j

∑ ^{≥ Lk}

i≠ j

∑

Inflexion at I (in isolation): c = 0 generic behaviour in : ³

(Ricca & Moffatt 1992)

Reidemeister type I move in action:

(TRACE mission 2002)

F _m F

∝ c ˆn

Lorentz force

inflexion

I X s, t ( ) ^{= s − 2}

3 t

s

, −ts

, s

Inflexional magnetic knots

From inflexion-free knots to magnetic braids

Definition. A knot in an inflexion-free configuration is a spiral knot.

Theorem (Ricca 2005). Let denote a loose magnetic knot in inflexional state. Then is in inflexional disequilibrium. K

K

Corollary. If is in inflexional disequilibrium, it relaxes to a spiral knot , for K t >t ^t

₀ .

K ^t

inflexional

instability

From minimal knots to spiral knots

Knots and links tabulation

10-crossing knot table (Tait 1885)

3-component links by KnotPlot

(Scharein 2000)

…

up to arbitrarily large # crossings

O(10

)

Groundstate energy of zero-framed knots and links

Relaxation of 0-framed, 5-crossing knots:

M

∝ c

Under signature-preserving flows, we have:

V = 1, Φ = 1, 0-framing c

# knot types

3 1

4 1

5 2

6 3

7 7

8 21

9 49

10 166

… …

ϕ K (

)

ϕ K

(

)

6

6

0

magnetic field:

B = 0, 1 L

dΦ

dr , 1 2 π r

dΦ

dr + 0, ∂ψ

∂ s , − ∂ψ

∂ϑ

twist parameter:

fluxes , :

Φ

Φ

Constrained minimization of magnetic energy of knots

B = 0, B (

( ) r ^{, B}

( ) ^r )

Φ

Φ

tubular knot : ; Mercier (orthogonal) system:

Theorem (Maggioni & Ricca 2009). Let us assume that (i) { V, Φ } invariant ( V = 1, Φ = 1 );

(ii) circular cross-section independent of arc-length;

(iii) independent of arc-length;

(iv) L independent of internal twist.

Then, we have ψ ˜

.

K V K ( ) ^{= π a}

^L ( ^r,ϑ,s )

h = Φ

/ Φ

∇ ⋅ B = 0

( )

R

L

ropelength: λ = L ^* /R ^* M

( ) h ^{= λ}

4/3

2 π

⁺

π

^h

λ

^{= m(} λ ^{, h)}

c _min

h

3 4 ⁵

6 7

8 9

10 Groundstate energy spectrum: averaging over complexity

m( λ , h)

c _min

m

( ) h ^{= 3}

2 π

^h

( h ≥ 2 )

(see also Chui & Moffatt 1992)

tightening: V = 1, Φ = 1 , h = 0

SONO ( Pieranski et al. 1998)

RIDGERUNNER (Ashton et al. 2010)

Groundstate energy spectrum of first 250 prime knots

10 crossings

9 crossings

8 crossings

7 crossings

6 crossings

5 crossings 4 crossings 3 crossings

m(#

) = m ( λ ^(#

^{), 0} )

M

= λ ^(#

⁾ 2 π

4/3

m(#

)

#

M

= 2 ( π

)

V = 1, Φ = 1 , h = 0 tight unknot:

(Rolfsen table)

m(#

) = 4.5ln #

+10.5

best fit:

linear fit over c

-families

Tight knots

RIDGERUNNER

m(#

)

m(#

) = m ( λ ^(#

^{), 0} )

M

= λ (#

) 2 π

4/3

M

= 2 ( π

)

V = 1, Φ = 1 , h = 0 tight unknot:

Knot energy spectrum by increasing ropelength (Ricca & Maggioni 2014)

#

RIDGERUNNER

(increasing ropelength)

m(#

)

m(#

) = m ( λ ^(#

^{), 0} )

M

= λ ^(#

⁾ 2 π

4/3

m(#

) = 4.5ln #

+ 9.3

best fit:

linear fit over c

-families

Tight links M

= 2 ( π

)

V = 1, Φ = 1 , h = 0 tight unknot:

Link energy spectrum by increasing ropelength (Ricca & Maggioni 2014)

#

(increasing ropelength)

RIDGERUNNER

V = 1, h = 0

Bending energy spectrum of elastic knots

e(#

) e(#

)

e(#

) = 0.2 ln #

+ 0.6

best fit:

linear fit over c

-families

Tight knots

RIDGERUNNER

#

RIDGERUNNER

e = E

E

= ∫

[ c(s) ]

^ds

2

π

E

= π ^R

²

π

tight torus:

(increasing ropelength)

e = E

E

= ∫

[ c(s) ]

^ds

2

π

E

= π ^R

²

π

V = 1, h = 0 tight torus:

Bending energy spectrum of elastic links

e(#

) e(#

)

e(#

) = 0.2 ln #

+ 0.5

best fit:

linear fit over c

-families

Tight links

RIDGERUNNER RIDGERUNNER

#

(increasing ropelength)

Magnetic versus bending energy (Ricca & Maggioni 2017)

#

(increasing ropelength)

χ ^(#

⁾

χ ^(#

⁾

8_4_3

Tight knots: χ ^(#

^{) = 21.62}

Tight links: χ ^(#

^{) = 21.42}

λ ^(#

⁾

[ ]

^{∝ a ln #}

^{+ b} [ ^{λ (#}

⁾ ]

^{∝ c}

in partial agreement with

O c ( )

^{≤ λ (#}

^{) ≤ O c} (

^ln

^c

)

(Buck & Simon 1999; Cantarella et al . 2002; Diao 2003; Diao et al. 2013).

m c (

) ^{≡ M}

m

= c

π

λ ( ) ^#

^{≥ 2 π}

^c

^{≈ 2.66 c}

; then , or

From lower bounds on energy (V = 1, Φ = 1 ), we have:

M

= 2 π

1/3

c

and ; since

∀ c

better than Buck & Simon (1999), where constant (4π/11) ^3/4 ≈ 1.10 . then

, .

,

m(#

) ∝ [ λ ^(#

⁾ ]

Assumption: , then from we have

m # ( )

^{≥ m c} (

) ^{= c}

π ^,

λ ^(# _K ) ∝ c _min ^3/4

# _K ~ A ^c

Ropelength versus topological crossing number