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Scott TaylorCarleton CollegeJanuary 16, 2011
When is a knot not knotted?
http
://w
ww.
mat
h.su
nysb
.edu
/~ja
ck/
John Milnorb. 19311962 Fields Medal1989 Wolf Prize2011 Abel Prize
Harold Tucker
Not
ices
, (42
) 10
Harold Tucker
Not
ices
, (42
) 10
Karol BorsukM
acTu
tor
Harold Tucker
Not
ices
, (42
) 10
Karol BorsukM
acTu
tor
Ralph FoxM
acTu
tor
The way math is done at Princeton.
cine
plex
.com
The total curvature of a plane curve is at least 2π.
Fenchel’s Theorem
The total curvature of a knotted curve in R3 is at least 4π.
Borsuk’s Conjecture (1947)
Solved by Fáry (1949) and Milnor (1950)
C(t) =
x(t)y(t)z(t)
Curvature
.
C(t) =
x(t)y(t)z(t)
Curvature
C�(t) =
x�(t)y�(t)z�(t)
.
C(t) =
x(t)y(t)z(t)
Curvature
C�(t) =
x�(t)y�(t)z�(t)
κ(t) = |C��(t)||C�(t)| = 1If then curvature is defined to be
.
κ(C) =
� b
aκ(t) dt
Total Curvature
The total curvature of a smooth curve
C(t) a ≤ t ≤ b
is defined to be
C(t) =
�R cos(t/R)R sin(t/R)
�0 ≤ t ≤ 2πR
R
An example
C(t) =
�R cos(t/R)R sin(t/R)
�0 ≤ t ≤ 2πR
C�(t) =
�− sin(t/R)cos(t/R)
�
R
An example
C(t) =
�R cos(t/R)R sin(t/R)
�0 ≤ t ≤ 2πR
C�(t) =
�− sin(t/R)cos(t/R)
�
C��(t) =1
R
�− cos(t/R)− sin(t/R)
�R
An example
C(t) =
�R cos(t/R)R sin(t/R)
�0 ≤ t ≤ 2πR
C�(t) =
�− sin(t/R)cos(t/R)
�
C��(t) =1
R
�− cos(t/R)− sin(t/R)
�
κ(t) = |C��(t)| = 1/R
R
An example
C(t) =
�R cos(t/R)R sin(t/R)
�0 ≤ t ≤ 2πR
C�(t) =
�− sin(t/R)cos(t/R)
�
C��(t) =1
R
�− cos(t/R)− sin(t/R)
�
κ(t) = |C��(t)| = 1/R
R
An example
κ(C) =
� 2πR
0
1
Rdt = 2πTotal Curvature:
The total curvature of a plane curve is at least 2π.
Fenchel’s Theorem
The total curvature of a knotted curve in R3 is at least 4π.
Borsuk’s Conjecture
Knottedness
A (smooth or polygonal) closed curve in R3 is knotted if it cannot be deformed into the unit circle in the xy-plane.
unknotted knotted (?)
Knottedness
A (smooth or polygonal) closed curve in R3 is knotted if it cannot be deformed into the unit circle in the xy-plane.
unknotted knotted (?)
When is a knot not knotted?
Detecting unknottedness
Theorem: If a closed curve has a single maximum (along some axis), then it is unknotted.
Detecting unknottedness
Theorem: If a closed curve has a single maximum (along some axis), then it is unknotted.
Bridge number in a direction X
Let X be a unit vector in R3. The bridge number of C in the direction X is the total number of maxima of the function C(t)⋅X. Denote it by b(X).
Bridge number in a direction X
Let X be a unit vector in R3. The bridge number of C in the direction X is the total number of maxima of the function C(t)⋅X. Denote it by b(X).
Bridge number in a direction X
Let X be a unit vector in R3. The bridge number of C in the direction X is the total number of maxima of the function C(t)⋅X. Denote it by b(X).
i
j
Bridge number in a direction X
Let X be a unit vector in R3. The bridge number of C in the direction X is the total number of maxima of the function C(t)⋅X. Denote it by b(X).
i
j
b(j) = 4
Bridge number in a direction X
Let X be a unit vector in R3. The bridge number of C in the direction X is the total number of maxima of the function C(t)⋅X. Denote it by b(X).
i
j
b(j) = 4b(i) = 2
Bridge number in a direction X
Let X be a unit vector in R3. The bridge number of C in the direction X is the total number of maxima of the function C(t)⋅X. Denote it by b(X).
i
j
b(j) = 4b(i) = 2
Note:
•Some directions may be degenerate, •b(X) also equals the number of minima.
�
|X|=1b(X) dA = 2κ(C).
The genius
Theorem (Milnor): If C is a smooth closed curve in R3, then:
2κ(C) =�|X|=1 b(X) dA
≥ 2�|X|=1 dA
= 8π.
κ(C) ≥ 4π.
The total curvature of a knotted curve in R3 is at least 4π.
Borsuk’s Conjecture proof:
Assume C is knotted. Then b(X) ≥ 2 for all X. Hence, by Milnor’s theorem:
Consequently,
�
|X|=1b(X) dA = 2κ(C).
Theorem (Milnor): If C is a smooth closed curve in R3, then:
�
|X|=1b(X) dA = 2κ(C).
Theorem (Milnor): If C is a smooth closed curve in R3, then:
proof:
1. Convert to polygonal curves.
2. Prove theorem for polygonal curves.
3.Prove that the polygonal theorem implies the smooth theorem.
Polygonal Approximations
Polygonal Approximations
Polygonal Approximations
vj
vj-1
vj+1
Polygonal Curvature
vj
vj-1
vj+1
Polygonal Curvature
θj exterior angle
κ(C) =�
j
θj
vj
vj-1
vj+1
Polygonal Curvature
θj exterior angle
2π/3
κ(C) = 2π
2π/3
2π/3
Polygonal Curvature
Theorem (Milnor):Let C be a polygonal curve in R3. Then�
|X|=1b(X) dA = 2κ(C).
�
|X|=1b(X) dA = 2κ(C).To Prove:
Note: If X is nondegenerate:
•the maxima and minima of C⋅X are at vertices. •varying X slightly doesn’t change b(X).
�
|X|=1b(X) dA = 2κ(C).To Prove:
Note: If X is nondegenerate:
•the maxima and minima of C⋅X are at vertices. •varying X slightly doesn’t change b(X).
Thus: the sphere of directions {X : |X| = 1} is
divided into finitely many regions where b(X) is constant.
�
|X|=1b(X) dA = 2κ(C).To Prove:
Note: If X is nondegenerate:
•the maxima and minima of C⋅X are at vertices. •varying X slightly doesn’t change b(X).
Thus: the sphere of directions {X : |X| = 1} is
divided into finitely many regions where b(X) is constant.
Define:
Areaj = Area of { X| vj is a max/min of C(t) ⋅X}.
�
|X|=1b(X) dA = 2κ(C).To Prove:
Areaj = Area on sphere where vj is a max/min.
Consequently:
2
�
|X|=1b(X) dA =
�
j
Areaj
What is Areaj?
What is Areaj?
What is Areaj?
What is Areaj?
What is Areaj?
What is Areaj?
What is Areaj?
Areaj = 2 · 4π · θj2π
Areaj = 4θj .
Putting it all together:
Since:
.
Areaj = 4θj .
Putting it all together:
Since:
2
�
|X|=1b(X) dA =
�
j
Areaj
We have:
.
Areaj = 4θj .
Putting it all together:
Since:
2
�
|X|=1b(X) dA =
�
j
Areaj
We have:
.
= 4�
j
θj
Areaj = 4θj .
= 4κ(C).
Putting it all together:
Since:
2
�
|X|=1b(X) dA =
�
j
Areaj
We have:
.
= 4�
j
θj
Areaj = 4θj .
= 4κ(C).
Putting it all together:
Since:
2
�
|X|=1b(X) dA =
�
j
Areaj
We have:
.
= 4�
j
θj
�
|X|=1b(X) dA = 2κ(C)
Consequently,
Dirk
Fer
us, 1
980,
Obe
rwol
fach
John Milnor
The ideas live on...
• Topology and Geometry are intricately related in low dimensions.• Bridge number is an important invariant in modern knot theory.• Total curvature plays an important role in applications of knot theory to chemistry.• The interplay between the continuous and the discrete is a prevalent theme in modern mathematics.
http://www.math.sunysb.edu/~jack/OSLO/PHOTOS/ORIG/cmonu.jpg
John Milnor lives on...
http://www.m
ath.su
nysb
.edu
/~jack
/OSL
O/PHOTO
S/ORIG
/jack
-abe
l7.jp
g
John Milnor
Sources and References
“On the total curvature of knots” J.W. Milnor. Annals 1950.
“A brief report on John Milnor’s brief excursions into differential geometry” M. Spivak. Topological Methods in Modern Mathematics. Publish or Perish, Inc. 1993.
“Curves of Finite Total Curvature” J. Sullivan. Discrete Differential Geometry. Birkhäuser. preprint: arXiv 2007.
What is Area(Vi)?
κ(C) =�
j
θj
θj
vj
vj-1
vj+1