Linking Integrals in Hyperspheres
Clayton Shonkwiler
(Joint with David Shea Vela-Vick)
Department of Mathematics
University of Pennsylvania
Sewanee
October 24, 2008
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A Catenane
Linking Number
Definition
Let K and L be smooth, oriented closed curves embedded in R3 .
Then the linking number Lk(K , L) between K and L is the (signed)
number of intersections of K with a surface bounded by L.
Linking Number
Definition
Let K and L be smooth, oriented closed curves embedded in R3 .
Then the linking number Lk(K , L) between K and L is the (signed)
number of intersections of K with a surface bounded by L.
Linking Number
Linking Number
Linking Number
Definition
Let K and L be smooth, oriented closed curves embedded in R3 .
Then the linking number Lk(K , L) between K and L is the (signed)
number of intersections of K with a surface bounded by L.
Linking Number
Definition
Let K and L be smooth, oriented closed curves embedded in R3 .
Then the linking number Lk(K , L) between K and L is the (signed)
number of intersections of K with a surface bounded by L.
Linking Number
Definition
Let K and L be smooth, oriented closed curves embedded in R3 .
Then the linking number Lk(K , L) between K and L is the (signed)
number of intersections of K with a surface bounded by L.
Linking Number
Definition
Let K and L be smooth, oriented closed curves embedded in R3 .
Then the linking number Lk(K , L) between K and L is the (signed)
number of intersections of K with a surface bounded by L.
Linking Number
Definition
Let K and L be smooth, oriented closed curves embedded in R3 .
Then the linking number Lk(K , L) between K and L is the (signed)
number of intersections of K with a surface bounded by L.
Linking Number
Definition
Let K and L be smooth, oriented closed curves embedded in R3 .
Then the linking number Lk(K , L) between K and L is the (signed)
number of intersections of K with a surface bounded by L.
The Gauß Linking Integral
Theorem (Gauß, 1833)
Let K = {x(s)} and L = {y(t)} be disjoint, smooth, oriented
closed curves embedded in R3 . Then the linking number between
K and L is given by
Z
x(s) − y(t)
1
dx dy
×
ds dt
Lk(K , L) =
·
4π K ×L |x(s) − y(t)|3
ds
dt
The Gauß Linking Integral
Theorem (Gauß, 1833)
Let K = {x(s)} and L = {y(t)} be disjoint, smooth, oriented
closed curves embedded in R3 . Then the linking number between
K and L is given by
Z
x(s) − y(t)
1
dx dy
×
ds dt
Lk(K , L) =
·
4π K ×L |x(s) − y(t)|3
ds
dt
dy
dt
x−
y
dx
ds
The “Visual” Proof of the Gauß Linking Integral
Define f : K × L → S 2 by
f (s, t) =
x(s) − y(t)
.
|x(s) − y(t)|
The “Visual” Proof of the Gauß Linking Integral
Define f : K × L → S 2 by
f (s, t) =
x(s) − y(t)
.
|x(s) − y(t)|
L
x−
y
K
The “Visual” Proof of the Gauß Linking Integral
Define f : K × L → S 2 by
f (s, t) =
x(s) − y(t)
.
|x(s) − y(t)|
K
L
Lk(K , L) = number of times f (s, t) is the north pole.
Covering the Sphere
Linking Number as Area
1
Lk(K , L) =
4π
Z
K ×L
x(s) − y(t)
·
|x(s) − y(t)|3
dx dy
×
ds
dt
ds dt
Linking Number as Area
1
Lk(K , L) =
4π
Area of S 2
Z
K ×L
x(s) − y(t)
·
|x(s) − y(t)|3
dx dy
×
ds
dt
ds dt
Linking Number as Area
1
Lk(K , L) =
4π
Z
K ×L
x(s) − y(t)
·
|x(s) − y(t)|3
dx dy
×
ds
dt
ds dt
Area of S 2

1
det 
|x(s) − y(t)|3
x1 − y1 x2 − y2 x 3 − y3
dx1
ds
dy1
dt
dx2
ds
dy2
dt
dx3
ds
dy3
dt


Linking Number as Area
1
Lk(K , L) =
4π
Z
K ×L
x(s) − y(t)
·
|x(s) − y(t)|3
dx dy
×
ds
dt
Note
The integrand is invariant under rigid motions of R3 .
ds dt
Our Goal
Goal
Find analogous linking integrals in higher dimensions and in
non-Euclidean spaces.
Higher-Dimensional Euclidean Spaces
In higher-dimensional Euclidean spaces, the same idea yields an
equivalent linking integral.
Higher-Dimensional Euclidean Spaces
In higher-dimensional Euclidean spaces, the same idea yields an
equivalent linking integral.
Theorem
If K = {x(s1 , s2 )} is a closed surface and L = {y(t)} is a closed
curve, both living in R4 , then
Z
1
−1
[x − y, dx, dy]
Lk(K , L) = 2
2π K ×L |x − y|4
Higher-Dimensional Euclidean Spaces
In higher-dimensional Euclidean spaces, the same idea yields an
equivalent linking integral.
Theorem
If K = {x(s1 , s2 )} is a closed surface and L = {y(t)} is a closed
curve, both living in R4 , then
Z
1
−1
[x − y, dx, dy]
Lk(K , L) = 2
2π K ×L |x − y|4
What about spheres?
The 3-sphere
The 3-sphere S 3 is defined as:
S 3 := {(x1 , x2 , x3 , x4 ) ∈ R4 : x12 + x22 + x32 + x42 = 1}.
Visualizing the 3-sphere
Intersection with a (hyper)plane:
Visualizing the 3-sphere
Intersection with a (hyper)plane:
Visualizing the 3-sphere
Intersection with a (hyper)plane:
Visualizing the 3-sphere
Stereographic projection:
Visualizing the 3-sphere
Stereographic projection:
Visualizing the 3-sphere
Gluing:
Visualizing the 3-sphere
Gluing:
Visualizing the 3-sphere
Gluing:
Visualizing the 3-sphere
Gluing:
Visualizing the 3-sphere
Gluing:
Visualizing the 3-sphere
Gluing:
Living on the 3-sphere
If you stand at the north pole and your friend moves away from
you, what do you see?
Living on the 3-sphere
If you stand at the north pole and your friend moves away from
you, what do you see?
Living on the 3-sphere
If you stand at the north pole and your friend moves away from
you, what do you see?
Living on the 3-sphere
If you stand at the north pole and your friend moves away from
you, what do you see?
A Linking Integral for the 3-sphere
Given K = {x(s)} and L = {y(t)}, two closed curves in the
3-sphere, we want to find an integral formula for Lk(K , L).
A Linking Integral for the 3-sphere
Given K = {x(s)} and L = {y(t)}, two closed curves in the
3-sphere, we want to find an integral formula for Lk(K , L).
K
K
L
The Key Idea
K
K
L
The Key Idea
CK
K
K
L
L
Bumping up the dimension of K
We bump up the dimension of K by taking the half-infinite cone
from the origin:
CK
CK := {τ x | x ∈ K , τ ∈ [0, ∞)}
L
Bumping up the dimension of K
We bump up the dimension of K by taking the half-infinite cone
from the origin:
CK
CK := {τ x | x ∈ K , τ ∈ [0, ∞)}
L
Then LkS 3 (K , L) = LkR4 (CK , L), which we can compute using the
linking integral in Rn+1 .
The Linking Integral for S 3
1
Lk(K , L) = 2
2π
Z
CK ×L
−1
[τ x − y, d(τ x), dy]
|τ x − y|4
The Linking Integral for S 3
Z
1
−1
Lk(K , L) = 2
[τ x − y, d(τ x), dy]
2π CK ×L |τ x − y|4
Z ∞
Z
1
τ
= 2
dτ [x, dx, y, dy]
4
2π K ×L
τ =0 |τ x − y|
The Linking Integral for S 3
Z
1
−1
Lk(K , L) = 2
[τ x − y, d(τ x), dy]
2π CK ×L |τ x − y|4
Z ∞
Z
1
τ
= 2
dτ [x, dx, y, dy]
4
2π K ×L
τ =0 |τ x − y|
Z
Ω(α)
1
[x, dx, y, dy]
= 2
2π K ×L sin3 α
The Linking Integral for S 3
1
Lk(K , L) = 2
2π
Z
K ×L
Ω(α)
[x, dx, y, dy]
sin3 α
Here α(s, t) is the spherical distance from x(s) to y(t) and
Z π
Ω(α) =
sin(θ − α) sin θ dθ.
θ=α
Linking Integrals for Hyperspheres
The same techniques work for spheres of arbitrary dimension:
Theorem (Kuperberg, DeTurck–Gluck,
Kk,
–Vela-Vick)
L`
Let
be disjoint, oriented, closed submanifolds of S n with
k + ` = n − 1. Then
Z
Ωk,` (α)
1
[x, dx, y , dy ]
Lk(K , L) =
n
Vol S K ×L sinn α
where
Z
π
Ωk,` (α) =
sink (θ − α) sin` θdθ
θ=α
and α(x, y ) is the geodesic distance in S n from x ∈ K to y ∈ L.