Quantum Topology

Quantum Topology

Born around/after 1984

Term (probably) coined by Louis Kauffman

What is it about?

What is it about?

What is it not about?

What is it not about?

How is it different from ”classical” topology?

How is it different from ”classical” topology?

specialized in Dim = 3:

3-dim Manifolds

Knots & Links in 3-Space

2-dim Surfaces

Mapping Class Groups

of

for these 3-dim Objects

Invariants “nicely” behaved under topological operations

Underlying Algebra specially “designed” for dim=3

Quantum Topology

Quantum Topology

What is it about?

What is it about?

Quantized (non-commutative) topological Spaces

Heuristic or physicist's mathematics

Insulated Combinatorics

But, the algebras used to find invariants, representations, and functors are quantizations or “classical” algebras

But, constructions inspired by intriguing relations to quatum field theories (CFTs & TQFT's) and integrable models.

Evolved highly structured & rigorous field of mathematics

Disproving the Ponicare Conjecture

But, relates to 3 & 4 dim topology, gauge theory, representation theory, group theory, operator algebra, geometry, ...

Quantum Topology

Quantum Topology

What is it not about?

What is it not about?

Invariants:

Top Alg

Top Alg

Classical

Quantum

Geometrical Constructions

Combinatorial Constructions

(p,H,t, ...)

(from combin. “presentations”)

General Dimensions

Tailored to dimension 3

Computation “geometrical”

Computation “algorithmic”

Content by construction

Numerical Invariants dominant

Quantum Topology

Quantum Topology

How is it different from ”classical” topology?

How is it different from ”classical” topology?

Example:

Isotopy Classes of

Knots and Links

Laurent Polynomials

(mod units)

Quantum Topology

Quantum Topology

Trefoil Projections

53-Torus Knot Projections

R1)

Reidemeister Moves

R2)

R3)

R1)

R2)

R3)

Braid

(Artin)

Relations

Reidemeister Moves

THEOREM

Two (generic) planar diagrams, say D and F,

represent isotopic knots/links if and only of they are equivalent under the moves R1-R3 (and planar isotopies)

I.e., D ~ D' ~ D'' ~ ... ~ D * ~ F

R#

R#

R#

R#

R#

Invariants

Function on isotopy classes of knots/links

=

Function on planar diagrams,

invariant under R1-R3

Reidemeister Moves

Kauffman Bracket

Kauffman Bracket

D

Assigns to planar diagram

Laurent Polynomial

Rules :

< >

< >

< >

-1

q

q

=

+

Z

[q,q ]

-1

'

< D >

< >

D

=

(-q - q )

2

-2

D

< >

< >

=

1

Kauffman Bracket

Kauffman Bracket

THEOREM

The bracket is invariant under R2 and R3.

D

< >

Moreover, it behaves under R1 as follows:

- q

-3

<... >

=

<... >

Note:

Z

Z

[q,q ]

-1

+q

-

Have Knot/Link Invariant in

Writhe Renormalization Jones Polynomial

Kauffman Bracket

Kauffman Bracket

Lemma

Notation

For a knot

K

K

denote by

knot obtained by plane reflection

For planar diagram

D

D

denote by

obtained by switching over and under crossings

the diagram

For Laurent Polynomial

P

P

denote by

polynomial

obtained by subtitution

q

-1

q

Lemma

K

is a projection of

K

If

D

then

D

is a projection of

Lemma

D

D

< > = < >

< > =

- q

4

- q

-4

Kauffman Bracket

Kauffman Bracket

< >=

For the right-handed Trefoil

T =

< > + < >

q

-1

q

- q )

(- q

4

-4

q

q

-1

(-q )

-3

2

= +

=

- q - q + q

5

-3

-7

- q - q + q

< >=

-5

3

7

Kauffman Bracket

Kauffman Bracket

< >

T

< >

T

=

- q - q + q

-5

3

7

- q - q + q

5

-3

-7

=

2

1+ q - q

8

12

1- q - q

4

12

-q

Hence

Corollary

=

The trefoil knot is NOT amphicieral

Z

[q,q ]

-1

Z

+q

-

and

are different in

< >

T

< >

T

Kauffman Bracket

Kauffman Bracket

Definition

D

X

For planar diagram

D

let

= {1,...,N}

set of crossings.

A state is a map

s:

D

X

{-1, +1}

D

s

= resolution (diagram w/o Xings) obtained by replacements:

n

s(n)= -1

if

if

s(n)= +1

w(s) =

S

n

s(n)

Functions on

D

X

{-1, +1}

C(s) =

Number of components of

D

s

Kauffman Bracket

Kauffman Bracket

s = [+1,-1,-1]

Example:

1

3

2

=

s

w(s) = -1

C(s) = 2

Definition

Lemma

State Sum

< | >

D

s

= q (-q +q )

w(s)

C(s)-1

2

-2

< >

D

S

=

D

X

s in

< | >

D

s

Kauffman Bracket

Kauffman Bracket

Top and Bottom Degrees of Polynomials

What happens if switch a +1 to -1 in one place a ?

s(a) = 1, s'(a) = -1 and s(n) = s'(n) for n=a

w(s') = w(s) -2

C(s') = C(s) + 1

Lemma

unchanged if A and B connected.

a

a

A

B

drop by -4 if A and B different components.

Top

= w(s) + 2 C(s) - 2

*

< | >

D

s

Bot

= w(s) - 2 C(s) + 2

*

< | >

D

s

Top

< | >

D

s

Kauffman Bracket

Kauffman Bracket

Strict maximum if components in pD not “self-touching”.

In this case:

D

X

'

constant p(n) = +1

Denote by p

Otherwise <

=

Similarly:

Top

>

=

Top

< | >

D

s

< | >

D

p

=

Bot

Bot

=

= -N - 2 C(-p) + 2

*

( > )

=

< >

D

< | >

D

p

=

< >

D

Top

=

= N + 2 C(p) - 2

*

< | >

D

p

Top

Kauffman Bracket

Kauffman Bracket

Alternating Knots and Links:

Admits a projection in which an overcrossing is always followed by an undercrossing (vice-versa)

D

p

?

Kauffman Bracket

Kauffman Bracket

Alternating Knots and Links:

D

p

No areas touch themselves

for “reduced” diagrams

< >

D

Top

= N + 2 C(p) - 2

*

C(p) =

# of red patches

Kauffman Bracket

Kauffman Bracket

Similarly ...

D

p

-

< >

D

Bot

= -N - 2 C(-p) + 2

*

C(-p) =

# of green patches

Alternating Knots and Links:

Kauffman Bracket

Kauffman Bracket

Knot Invariant:

< >

D

Span

< >

D

Top

< >

D

Bot

=

-

= 2N + 2 (C(p)+C(-p)) - 4

= 2(N+F-2)

= 2E

= 4N

Alternating Knots and Links:

Kauffman Bracket

Kauffman Bracket

Theorem

< >

D

Span

<

=

4N

for general knots/links

Corollary

(Tait Conjecture 1898)

Suppose a knot/link has a presentation by

an alternating, reduced planar diagram.

Then there is NO presentation of the knot

with fewer crossings.

< >

Span

=

4N

for reduced, alternating knots/links

D

Alternating Knots and Links:

Thomas Kerler

2005-01-23T03:46:55