Generalized Gauss maps and triple linking
integrals
Clayton Shonkwiler
University of Georgia
Isaac Newton Institute
International Centre for Mathematical Sciences
Edinburgh
October 18, 2012
Joint work with Dennis DeTurck, Herman Gluck, Rafal
Komendarczyk, Paul Melvin, and David Shea Vela-Vick
+ Haggai Nuchi.
Motivation
To find topological invariants of vector fields which provide lower
bounds on the field energy.
The model
Definition
The helicity of a vector field V supported on a compact domain
Ω ⊂ R3 is
Z
1
x−y
H(V ) :=
dx dy.
V (x) × V (y) ·
4π Ω×Ω
|x − y|3
A lower bound for energy
Theorem (Arnol0 d, . . . )
|H(V )| ≤ λ(Ω)E (V ),
where λ(Ω) is a positive constant depending on the shape and size
of Ω.
Theorem (Woltjer, Moffatt, Arnol0 d, . . . )
H(V ) is invariant under any volume-preserving diffeomorphism
which is homotopic to the identity.
The dream
Arnol0 d and Khesin:
The dream is to define such a hierarchy of invariants for
generic vector fields such that, whereas all the invariants
of order ≤ k have zero value for a given field and there
exists a nonzero invariant of order k + 1, this nonzero
invariant provides a lower bound for the field energy.
Where does helicity come from?
1
H(V ) =
4π
Z
V (x) × V (y) ·
Ω×Ω
x−y
dx dy.
|x − y|3
Where does helicity come from?
1
H(V ) =
4π
Z
V (x) × V (y) ·
Ω×Ω
x−y
dx dy.
|x − y|3
If X ∪ Y is a two-component link in R3 such that
X = {x(s) : s ∈ S 1 } and Y = {y(t) : t ∈ S 1 }, then the linking
number between X and Y is given by the Gauss linking integral
1
Lk(X , Y ) =
4π
Z
S 1 ×S 1
dx dy x(s) − y(t)
×
·
ds dt.
ds
dt |x(s) − y(t)|3
Where does helicity come from?
1
H(V ) =
4π
Z
1
Lk(X , Y ) =
4π
Z
dy
dt
V (x) × V (y) ·
Ω×Ω
S 1 ×S 1
x−
x−y
dx dy.
|x − y|3
dx dy x(s) − y(t)
×
·
ds dt.
ds
dt |x(s) − y(t)|3
y
dx
ds
Where does helicity come from?
1
H(V ) =
4π
Z
1
Lk(X , Y ) =
4π
Z
V (x) × V (y) ·
Ω×Ω
S 1 ×S 1
x−y
dx dy.
|x − y|3
dx dy x(s) − y(t)
×
·
ds dt.
ds
dt |x(s) − y(t)|3
The interpretation of the invariant in terms of
conservation of knottedness of magnetic lines of force
(which are frozen in the fluid) is immediate.
—Keith Moffatt
Geometric naturality
Lk(X , Y ) =
1
4π
Z
S 1 ×S 1
dx dy x(s) − y(t)
×
·
ds dt.
ds
dt |x(s) − y(t)|3
The integrand is invariant under the action of any
orientation-preserving isometry of R3 ; i.e., if h ∈ E + (3), then the
integrands for the links X ∪ Y and h(X ) ∪ h(Y ) are identical
functions of s and t.
The goal
The goal is to find “generalized Gauss linking integrals” which
compute higher-order linking invariants and are geometrically
natural (i.e. coordinate-independent) in the same sense.
The hope is that higher-order helicities can be defined by extending
these integrals to vector fields.
Whence the Gauss linking integral?
The Gauss map of a two-component link L = X ∪ Y is
gL : S 1 × S 1
(s, t)
- S2
- x(s) − y(t)
|x(s) − y(t)|
Whence the Gauss linking integral?
The Gauss map of a two-component link L = X ∪ Y is
gL : S 1 × S 1
(s, t)
- S2
- x(s) − y(t)
|x(s) − y(t)|
Theorem (Gauss)
The degree of gL is equal to the linking number of L.
Whence the Gauss linking integral?
The Gauss map of a two-component link L = X ∪ Y is
gL : S 1 × S 1
(s, t)
- S2
- x(s) − y(t)
|x(s) − y(t)|
Theorem (Gauss)
The degree of gL is equal to the linking number of L.
If ω is the area form on S 2 normalized to have area 1, then
Z
deg(gL ) =
gL∗ ω.
S 1 ×S 1
Expanding gives the Gauss linking integral.
Key Idea
Interpret invariants of links as homotopy invariants of maps.
Link homotopy
Definition
A link homotopy of a link L is a deformation during which each
component may cross itself, but distinct components must remain
disjoint.
!
n=3
Three-component links were classified up to link-homotopy by
Milnor:
• The pairwise linking numbers p, q, r .
• The triple linking number µ = µ123 , which is an integer
modulo gcd(p, q, r ).
Flux tubes
If V is supported on flux tubes, then µ provides a lower bound for
the field energy (Monastyrsky–Retakh, Berger, etc.; also
Freedman–He).
Defining a map
For x, y , z ∈ R3 which are all distinct, the following map F to
R3 − {~0} is well-defined:
z
n
γ
b
x
a
α
c
β
y
F (x, y, z) =
~b
~a
~c
+
+
~
|~a| |b| |~c |
!
+ (sin α + sin β + sin γ)~n.
The generalized Gauss map
Suppose L = X ∪ Y ∪ Z is a three component link in R3 with
X = {x(s) : s ∈ S 1 }
Y = {y(t) : t ∈ S 1 }
Z = {z(u) : u ∈ S 1 }.
The generalized Gauss map
Suppose L = X ∪ Y ∪ Z is a three component link in R3 with
X = {x(s) : s ∈ S 1 }
Y = {y(t) : t ∈ S 1 }
Z = {z(u) : u ∈ S 1 }.
Recall
F (x, y, z) =
~b
~a
~c
+
+
|~a| |~b| |~c |
!
+ (sin α + sin β + sin γ)~n.
The generalized Gauss map
Suppose L = X ∪ Y ∪ Z is a three component link in R3 with
X = {x(s) : s ∈ S 1 }
Y = {y(t) : t ∈ S 1 }
Z = {z(u) : u ∈ S 1 }.
Recall
F (x, y, z) =
~b
~a
~c
+
+
|~a| |~b| |~c |
!
+ (sin α + sin β + sin γ)~n.
For 3-component link L, define the generalized Gauss map
FL : S 1 × S 1 × S 1 → S 2 by
FL (s, t, u) :=
F (x(s), y(t), z(u))
.
|F (x(s), y(t), z(u))|
Pontryagin’s theorem
−→
Pontryagin’s theorem
−→
Pontryagin showed that classes in [S 1 × S 1 × S 1 , S 2 ] are
completely determined by:
• the degrees p, q, r on the 2-dimensional subtori and
• a “Hopf invariant” ν, which is an integer modulo
2 gcd(p, q, r ).
The triple linking number as a Pontryagin invariant
Theorem (with DeTurck, Gluck, Komendarczyk, Melvin, and
Vela-Vick)
The map L 7→ FL from link homotopy classes of three-component
links to [S 1 × S 1 × S 1 , S 2 ] is injective. In particular
• The degrees of the restrictions of FL to the 2-dimensional
subtori are equal to the pairwise linking numbers of L
• ν(FL ) ≡ 2µ(L) mod 2 gcd(p, q, r ).
Whitehead’s integral formula for the Hopf invariant
Whitehead’s integral formula for the Hopf invariant
• f : S 3 → S 2 smooth.
• ω the normalized area form on S 2 ( S 2 ω = 1).
R
Whitehead’s integral formula for the Hopf invariant
Whitehead’s integral formula for the Hopf invariant
• f : S 3 → S 2 smooth.
• ω the normalized area form on S 2 ( S 2 ω = 1).
• f ∗ ω is closed and, therefore, exact.
R
Whitehead’s integral formula for the Hopf invariant
Whitehead’s integral formula for the Hopf invariant
• f : S 3 → S 2 smooth.
• ω the normalized area form on S 2 ( S 2 ω = 1).
• f ∗ ω is closed and, therefore, exact.
R
• Then
Z
Hopf(f ) =
d −1 (f ∗ ω) ∧ f ∗ ω,
S3
where d −1 (f ∗ ω) is any primitive for f ∗ ω.
An integral formula for the µ invariant
If L has pairwise linking numbers all zero, then ωL = FL∗ ω is an
exact 2-form on S 1 × S 1 × S 1 .
An integral formula for the µ invariant
If L has pairwise linking numbers all zero, then ωL = FL∗ ω is an
exact 2-form on S 1 × S 1 × S 1 .
Whitehead’s integral transplanted to S 1 × S 1 × S 1 :
Z
ν(FL ) =
d −1 (ωL ) ∧ ωL
(S 1 )3
for any primitive d −1 (ωL ) of ωL .
An integral formula for the µ invariant
If L has pairwise linking numbers all zero, then ωL = FL∗ ω is an
exact 2-form on S 1 × S 1 × S 1 .
Whitehead’s integral transplanted to S 1 × S 1 × S 1 :
Z
ν(FL ) =
d −1 (ωL ) ∧ ωL
(S 1 )3
for any primitive d −1 (ωL ) of ωL .
Therefore, by the above theorem,
1
1
µ(L) = ν(FL ) =
2
2
Z
(S 1 )3
d −1 (ωL ) ∧ ωL .
Inverting the curl
1
µ(L) = ν(FL ) =
2
where ωL = FL∗ ω.
Z
S 1 ×S 1 ×S 1
d −1 (ωL ) ∧ ωL
Inverting the curl
1
µ(L) = ν(FL ) =
2
Z
d −1 (ωL ) ∧ ωL
S 1 ×S 1 ×S 1
where ωL = FL∗ ω.
The geometrically natural choice of d −1 (ωL ) is the primitive with
smallest energy, which can be obtained by convolving ωL with the
fundamental solution ϕ of the scalar Laplacian and taking the
exterior co-derivative:
d −1 (ωL ) = δ(ϕ ∗ ωL ).
The fundamental solution
Proposition
The fundamental solution of the scalar Laplacian on the 3-torus is
given by the Fourier series
ϕ(s, t, u) =
1
8π 3
X
(n1 ,n2 ,n3 )6=~0
e i(sn1 +tn2 +un3 )
.
n12 + n22 + n32
The fundamental solution
Proposition
The fundamental solution of the scalar Laplacian on the 3-torus is
given by the Fourier series
ϕ(s, t, u) =
1
8π 3
X
(n1 ,n2 ,n3 )6=~0
e i(sn1 +tn2 +un3 )
.
n12 + n22 + n32
Generalized Gauss integral
Theorem (with DeTurck, Gluck, Komendarczyk, Melvin,
Nuchi, and Vela-Vick)
For a three-component link L in R3 ,
1
µ(L) =
2
Z
δ(ϕ ∗ ωL ) ∧ ωL
S 1 ×S 1 ×S 1
= 8π 3
X
~n∈Z3 −{~0}
~a~n × ~b~n
~
· |~nn| ,
2
where ~a~n and ~b~n are the real and imaginary parts of the Fourier
coefficients of ωL .
The integrand is invariant under orientation-preserving isometries
of R3 .
An approximation
L1 (s) = (2 cos s, 7 sin s, 0)
L2 (t) = (0, 2 cos t, 7 sin t)
L3 (u) = (7 sin u, 0, 2 cos u)
Approximating Fourier coefficients by splitting the s, t, and u
intervals into 256 subintervals and for −64 ≤ n1 , n2 , n3 ≤ 64, yields
the approximation
µ(L) ' −0.99999997.
Future directions
• Does this integral for the triple linking number lead to a
higher helicity for vector fields on arbitrary domains (i.e., not
just on flux tubes or solid handlebodies)? Are there other
physically significant interpretations?
Future directions
• Does this integral for the triple linking number lead to a
higher helicity for vector fields on arbitrary domains (i.e., not
just on flux tubes or solid handlebodies)? Are there other
physically significant interpretations?
• Does this approach yield geometric integral formulas for other
link invariants?
Thanks!