Scott TaylorCarleton CollegeJanuary 16, 2011

When  is  a  knot  not  knotted?

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John  Milnorb.  19311962  Fields  Medal1989  Wolf  Prize2011  Abel  Prize

Harold  Tucker

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Harold  Tucker

Not

ices

, (42

) 10

Karol  BorsukM

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Harold  Tucker

Not

ices

, (42

) 10

Karol  BorsukM

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Ralph  FoxM

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The way math is done at Princeton.

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.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

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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