Lecture Notes for 120 - UCLA Department of Mathematics

3.3. CLOSED SPACE CURVES 69
Theorem 3.3.3. (Fary, 1949 and Milnor, 1950) If a simple closed space curve
is knotted, then
ˆ
appleds 4⇡
Proof. We assume that ´ appleds < 4⇡ and show that the curve is not knotted.
Crofton’s formula tells us that
ˆ
ˆ
1
n T (x) dx = appleds < 4⇡
4 S 2
As the sphere has area 4⇡ this can only happen if we can find x such that n T (x) apple 3.
Now observe that
d (q · x)
= T (s) · x
ds
So the function q · x has at most three critical points. Since q is closed there
will be a maximum and a minimum. The third critical point should it exist can
consequently only be an inflection point. Assume that the minimum is obtained at
s =0and the maximum at s 0 2 (0,L). The third critical point can be assumed to
be in (0,s 0 ). This implies that the function q (s) · x is strictly increasing on (0,s 0 )
and strictly decreasing on (s 0 ,L). For each t 2 (0,s 0 ) we can then find a unique
s (t) 2 (s 0 ,L) such that q (t) · x = q (s (t)) · x. Jointhetwopointsq (t) and q (s (t))
by a segment. These segments will sweep out an area whose boundary is the curve
and no two of the segments intersect as they belong to parallel planes orthogonal
to x. This shows that the curve is the unknot.
⇤
Exercises.
(1) Let q be a unit speed spherical curve
(a) Show that
apple 2 = 1+apple 2 g
N = 1 apple ( q + apple gS)
B = 1 apple (apple gq + S)
⌧ =
1
1+apple 2 g
(b) Show that q is planar if and only if the curvature is constant.
(2) Show that for a regular spherical curve q (t)
h
i
det q
dq d 2 q
dt dt
apple g =
2
ds
dt
(3) (Jacobi) Let q (s) :[0,L] ! R 3 be a closed unit speed curve with positive
curvature and consider the unit normal N as a closed curve on S 2 .
(a) Show that if s N denotes the arclength parameter of N, then
✓ dsN
dapple g
ds
3
◆ 2
= apple 2 q + ⌧ 2 q
ds
where apple q and ⌧ q are the curvature and torsion of q.