[PDF] - Mosaic Knots Mosaic Knots Quantum Knots & Quantum Knot PDF Document

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1

Quantum Quantum Knots ??? Knots ???

An Intuitive Overview An Intuitive Overview

f the Theory of
f the Theory of

Quantum Knots Quantum Knots Samuel Lomonaco Samuel Lomonaco

University of Maryland Baltimore County (UMBC) University of Maryland Baltimore County (UMBC) Email: Lomonaco@UMBC.edu Email: Lomonaco@UMBC.edu WebPage WebPage: www.csee.umbc.edu/~lomonaco : www.csee.umbc.edu/~lomonaco

L-

O

O-

O

O-

P

P This is work in collaboration with Louis Kauffman

Lecture II: Quantum Knots and Mosaics,
Lecture III: Quantum Knots & Lattices,
Lecture I: A Rosetta Stone for Quantum

Computing

Lecture IV: Intuitive Overview of the Theory
f Quantum Knots

PowerPoint Lectures and Exercises can be found at: www.csee.umbc.edu/~lomonaco Lomonaco and Kauffman, Lomonaco and Kauffman, Quantum Knots and Quantum Knots and Lattices, Lattices, to appear soon on quant to appear soon on quant-

ph

ph This talk is based on the paper: This talk is based on the paper: This talk was motivated by: This talk was motivated by: Lomonaco and Kauffman, Lomonaco and Kauffman, Quantum Knots and Quantum Knots and Mosaics, Mosaics, Journal of Quantum Information Journal of Quantum Information Processing, vol. 7, Nos. 2 Processing, vol. 7, Nos. 2-

3, (2008), 85

3, (2008), 85-

115.
115. An earlier version can be found at:

An earlier version can be found at: http://arxiv.org/abs/0805.0339 http://arxiv.org/abs/0805.0339

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Rasetti, Mario, and Tullio Regge, Rasetti, Mario, and Tullio Regge, Vortices in He II, Vortices in He II, current algebras and quantum knots, current algebras and quantum knots, Physica 80 A, Physica 80 A, North North-

Holland, (1975), 217

Holland, (1975), 217-

2333.

2333.

This talk was also motivated by: This talk was also motivated by:

Kitaev Kitaev, Alexei Yu, , Alexei Yu, Fault Fault-

tolerant quantum computation

tolerant quantum computation by by anyons anyons, , http://arxiv.org/abs/quant http://arxiv.org/abs/quant-

ph/9707021

ph/9707021 Lomonaco, Samuel J., Jr., Lomonaco, Samuel J., Jr., The modern legacies of The modern legacies of Thomson's atomic vortex theory in classical Thomson's atomic vortex theory in classical electrodynamics, electrodynamics, AMS PSAPM/51, Providence, RI AMS PSAPM/51, Providence, RI (1996), 145 (1996), 145 -

166.

166. Kauffman and Lomonaco, Kauffman and Lomonaco, Quantum Knots, Quantum Knots, SPIE Proc. SPIE Proc.

n Quantum Information & Computation II (ed. by
n Quantum Information & Computation II (ed. by

Donkor Donkor, , Pirich Pirich, & Brandt), (2004), 5436 , & Brandt), (2004), 5436-

30, 268

30, 268-

284.
284. http://xxx.lanl.gov/abs/quant

http://xxx.lanl.gov/abs/quant-

ph/0403228

ph/0403228

Throughout this talk: “Knot” means either a knot or a link

Preamble Preamble

Thinking Outside the Box Thinking Outside the Box

Knot Theory Knot Theory Quantum Mechanics Quantum Mechanics is a tool for exploring is a tool for exploring

Objectives Objectives

We seek to define a quantum knot in such

We seek to define a quantum knot in such a way as to represent the state of the a way as to represent the state of the knotted rope, i.e., the particular spatial knotted rope, i.e., the particular spatial configuration of the knot tied in the rope. configuration of the knot tied in the rope.

We also seek to model the ways of

We also seek to model the ways of moving the rope around (without cutting the moving the rope around (without cutting the rope, and without letting it pass through rope, and without letting it pass through itself.) itself.)

We seek to create a quantum system

We seek to create a quantum system that simulates a closed knotted physical that simulates a closed knotted physical piece of rope. piece of rope.

Rules of the Game Rules of the Game

Find a mathematical definition of a quantum Find a mathematical definition of a quantum knot that is knot that is

Physically meaningful, i.e., physically

Physically meaningful, i.e., physically implementable, and implementable, and

Simple enough to be workable and

Simple enough to be workable and useable. useable.

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Aspirations Aspirations

We would hope that this definition will be We would hope that this definition will be useful in modeling and predicting the useful in modeling and predicting the behavior of knotted vortices that actually behavior of knotted vortices that actually

ccur in quantum physics such as
ccur in quantum physics such as
In supercooled helium II

In supercooled helium II

In the Bose

In the Bose-

Einstein Condensate

Einstein Condensate

In the Electron fluid found within the

In the Electron fluid found within the fractional quantum Hall effect fractional quantum Hall effect Knot Theory Quantum Mechanics Group Representation Theory

=

Formal Rewriting System

=

Formal Rewriting System

=

Group Representation

Themes Themes Overview Overview

Preamble

Preamble

Mosaic Knots

Mosaic Knots

Quantum Mechanics: Whirlwind Tour

Quantum Mechanics: Whirlwind Tour

Preamble to Lattice Knots

Preamble to Lattice Knots

Lattice Knots

Lattice Knots

Quantum Knots & Quantum Knot Systems

Quantum Knots & Quantum Knot Systems

Q. Knots & Q. Knot Systems
Q. Knots & Q. Knot Systems
Future Directions & Open Questions

Future Directions & Open Questions via Mosaics via Mosaics via Lattices via Lattices

Mosaic Knots Mosaic Knots

Transforming Knot Theory into Transforming Knot Theory into a formal Rewriting System a formal Rewriting System Lomonaco and Kauffman, Lomonaco and Kauffman, Quantum Knots and Quantum Knots and Mosaics, Mosaics, Journal of Quantum Information Journal of Quantum Information Processing, vol. 7, Nos. 2 Processing, vol. 7, Nos. 2-

3, (2008), 85

3, (2008), 85-

115.
115. An earlier version can be found at:

An earlier version can be found at: http://arxiv.org/abs/0805.0339 http://arxiv.org/abs/0805.0339

Mosaic Mosaic Tiles Tiles

Let denote the following set of Let denote the following set of 11 11 symbols, called symbols, called mosaic mosaic (unoriented unoriented) ) tiles tiles: ( ) u

T

Please note that, up to rotation, there are Please note that, up to rotation, there are exactly exactly 5 tiles tiles

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Mosaic Knots Mosaic Knots

A 4-mosaic trefoil Figure Eight Knot Figure Eight Knot 5-

Mosaic

Mosaic

Hopf Link Hopf Link 4-

Mosaic

Mosaic Let K(n) = the set of n-mosaic knots

Sub Sub-

Mosaic Moves

Mosaic Moves

A Cut & Paste Move A Cut & Paste Move

1

P

← ← →

SLIDE 5

5

1

P

← ← → A Cut & Paste Move A Cut & Paste Move

1

P

← ← → A Cut & Paste Move A Cut & Paste Move

We will now re-express the standard moves on knot diagrams as sub-mosaic moves.

Planar Planar Isotopy Isotopy Moves Moves as as Sub Sub-

Mosaic Moves

Mosaic Moves

11 Planar Isotopy (PI) Moves on Mosaics 11 Planar Isotopy (PI) Moves on Mosaics

3

P

← ← →

4

P

← ← →

5

P

← ← →

7

P

← ← →

8

P

← ← →

9

P

← ← →

2

P

← ← →

1

P

← ← →

6

P

← ← →

10

P

← ← →

11

P

← ← → Planar Isotopy (PI) Moves on Mosaics Planar Isotopy (PI) Moves on Mosaics

It is understood that each of the above moves It is understood that each of the above moves depicts all moves obtained by rotating the depicts all moves obtained by rotating the sub sub-

mosaics by

mosaics by 0, , 90 90, , 180 180, or , or 270 270 degrees. degrees.

2 2 ×

For example, For example, represents each of the following represents each of the following 4 4 moves: moves:

1

P

← ← →

1

P

← ← →

1

P

← ← →

1

P

← ← →

1

P

← ← →

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Planar Isotopy (PI) Moves on Mosaics Planar Isotopy (PI) Moves on Mosaics

Each of the PI Each of the PI 2-submosaic moves represents submosaic moves represents any one of the any one of the (n (n-

2+1)

2+1)2 possible moves on an possible moves on an n-mosaic mosaic

Reidemeister Reidemeister Moves Moves as as Sub Sub-

Mosaic Moves

Mosaic Moves

Reidemeister (R) Moves on Mosaics Reidemeister (R) Moves on Mosaics

2

R

← ← →

1

R

← ← →

' 1

R

← ← →

' 2

R

← ← →

'' 2

R

← ← →

''' 2

R

← ←→ Reidemeister (R) Moves on Mosaics Reidemeister (R) Moves on Mosaics

3

R

← ← →

' 3

R

← ← →

''' 3

R

← ←→

'' 3

R

← ← →

( ) 3 iv

R

← ← →

( ) 3 v

R

← ← →

Each Each PI move PI move and each and each R move R move is a is a permutation permutation of the set of all

f the set of all

knot knot n-mosaics mosaics K(n)

PI & R Moves on Mosaics PI & R Moves on Mosaics

In fact, In fact, each PI and R move, each PI and R move, as a as a permutation, is a permutation, is a product product of

f disjoint

disjoint transpositions transpositions. We define the We define the ambient ambient group group as the as the subgroup of the group of all permutations subgroup of the group of all permutations

f the set
f the set K(n) generated by the

generated by the all PI all PI moves moves and and all all Reidemeister Reidemeister moves moves.

( ) A n The Ambient Group The Ambient Group ( ) A n

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Knot Type Knot Type

ι

 →

The Mosaic Injection The Mosaic Injection

( ) ( 1)

:

n n

M M ι

+

→

We define the We define the mosaic mosaic injection injection

( ) ( 1)

:

n n

M M ι

+

→

( ) , ( 1) ,

if 0 ,

therwise

n i j n i j

K i j n K

+

 ≤ < ≤ <  =    K(n+1) K(n) Mosaic Knot Type Mosaic Knot Type

'

n

K K ∼

provided there exists an element of the ambient provided there exists an element of the ambient group that transforms into . group that transforms into .

( ) A n K ' K

Def

Def. Two

Two n-mosaic knots mosaic knots and and are of are of the same the same knot knot n-type type, written , written

K ' K

'

n k k k

K i K ι

+

∼

Two Two n-mosaics and are of the same mosaics and are of the same knot knot type type if there exists a non if there exists a non-

negative

negative integer integer k such that such that

K ' K

1 2 2 3

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8

4 4 5 6 6 7 8 8 9

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9

10 10 11 11 12 12 13 13 14 14 15 15

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10

16 16 17 17

'

n

K K ∼ '

n

K K ∼

Conjecture: The Mosaic Knots formal rewriting system fully captures tame knot theory. Recently, Takahito Kuriya has proven this conjecture,

T.

T. Kuriya

Kuriya, , On a Lomonaco On a Lomonaco-

Kauffman Conjecture,

Kauffman Conjecture, arXiv:0811.0710. arXiv:0811.0710.

Quantum Quantum Mechanics Mechanics

Whirlwind Tour Whirlwind Tour

State of a Quantum System State of a Quantum System

The state of a Quantum System is a vector (pronounced ket ) in a Hilbert space . ψ H ψ

Def. A Hilbert Space is a vector

space over together with an inner product such that

H

,

: − − − − × → H H H H

1)

&

1 2 1 2

, , , u u v u v u v + = + = +

1 2 1 2

, , , vu u vu v u + = + = +

2)

, , u v u v λ λ λ λ =

3)

, , u v v u =

4) Cauchy seq in ,

∀

1 2

, , u u … H lim

n n

u

→∞ →∞

∈ H

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11

Dynamic Behavior of Q. Sys. Dynamic Behavior of Q. Sys.

The dynamic behavior of a quantum system is determined by Schroedinger’s equation. ( ) U t ψ ψ =

Schroedinger’s Equation

i H t ψ ψ ∂ = ∂

IN

OUT

where is time, and where is a curve in the group of unitary transformations on the state space . t ( ) U t ( ) U H U H H ψ

Initial State

H

Hamiltonian Dynamic State

Observable Observable An observable is a Hermitian operator

n the state space , i.e., a linear

transformation such that Ω

† T

Ω = Ω = Ω = Ω

H Quantum Measurement Quantum Measurement In In Out Out

j

λ

j j j

P P ψ ψ ψ ψ ψ ψ =

ψ

BlackBox BlackBox MacroWorld MacroWorld Quantum Quantum World World Eigenvalue Eigenvalue Observable Observable

Q. Sys.
Q. Sys.

State State

Q. Sys.
Q. Sys.

State State

P r

j

b

P ψ ψ ψ ψ =

j j j

P λ Ω = Ω = ∑

where where Spectral Decomposition Spectral Decomposition

Physical Physical Reality Reality Philosopher Philosopher Turf Turf

Ω

Quantum Knots Quantum Knots & Quantum Knot Systems Quantum Knot Systems

Recall that is the set of all n-mosaic knots.

The Hilbert Space The Hilbert Space

f Quantum Knots
f Quantum Knots

( ) n

K

For Q.M. systems, we need an underlying Hilbert space. So we define:

K(n)

The The Hilbert Hilbert space space

f
f quantum

quantum knots knots is the Hilbert space with the set of n-mosaic knots mosaic knots as its orthonormal basis, i.e., with orthonormal basis

( ) n

K K(n)

{ }

( )

:

n

K K K ∈ K(n)

+

K = 2 An Example of a Quantum Knot An Example of a Quantum Knot

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12

Since each element is a permutation, Since each element is a permutation, it is a linear transformation that simply it is a linear transformation that simply permutes basis elements. permutes basis elements.

( ) g A n ∈ The Ambient Group as a The Ambient Group as a Unitary Unitary Group Group ( ) A n

We We identify identify each element with the each element with the linear transformation defined by linear transformation defined by

( ) g A n ∈

( ) ( ) n n

K gK → K K K K

Hence, under this identification, the

Hence, under this identification, the ambient ambient group group becomes a becomes a discrete group discrete group of

f

unitary transfs unitary transfs on the Hilbert space .

n the Hilbert space .

( ) n

K ( ) A n

+

K = 2 An Example of the Group Action An Example of the Group Action

+

2

R K =

2

R

+

K = 2 ( ) A n The Quantum Knot System The Quantum Knot System (

) ( )

( ),

( )

n A n

K

( ) ( ) ( ) ( )

(1) ( ) ( 1)

, (1) , ( ) , ( 1)

n n

A A n A n

ι ι ι ι ι ι +

→ → → → → + → K K K K K

Physically

Physically Implementable Implementable

Def Def. A A quantum quantum knot knot system system is a is a quantum system having as its state space, quantum system having as its state space, and having the Ambient group as its set and having the Ambient group as its set

f accessible unitary transformations.
f accessible unitary transformations.

( ) n

K ( ) A n

( ) ( )

( ),

( )

n A n

K

Physically Physically Implementable Implementable Physically Physically Implementable Implementable

The states of quantum system are The states of quantum system are quantum quantum knots

knots. The elements of the ambient

. The elements of the ambient group are group are quantum quantum moves moves.

( ) A n

( ) ( )

( ),

( )

n A n

K The Quantum Knot System The Quantum Knot System (

) ( )

( ),

( )

n A n

K

( ) ( ) ( ) ( )

(1) ( ) ( 1)

, (1) , ( ) , ( 1)

n n

A A n A n

ι ι ι ι ι ι +

→ → → → → + → K K K K K

Physically

Physically Implementable Implementable Physically Physically Implementable Implementable Physically Physically Implementable Implementable

Choosing an integer Choosing an integer n n is analogous to is analogous to choosing a length of rope. The longer the choosing a length of rope. The longer the rope, the more knots that can be tied. rope, the more knots that can be tied. The parameters of the ambient group are The parameters of the ambient group are the the “knobs” “knobs” one turns to

ne turns to spacially

spacially manipulate manipulate the quantum knot. the quantum knot.

( ) A n Quantum Knot Type Quantum Knot Type

Def Def. Two quantum knots and are Two quantum knots and are

f the
f the same

same knot knot n-type type, written , written

1

K

2

K

1 2 , n

K K ∼

provided there is an element s.t. provided there is an element s.t.

( ) g A n ∈

1 2

g K K =

They are of the They are of the same same knot knot type type, written , written

1 2 ,

K K ∼

1 2 m m n m

K K ι ι

+

∼

provided there is an integer provided there is an integer such that such that

m ≥ Two Quantum Knots of the Same Knot Type Two Quantum Knots of the Same Knot Type

2

R

+

K = 2

+

K = 2

+

2

R K =

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13

Two Quantum Knots NOT of the Same Knot Type Two Quantum Knots NOT of the Same Knot Type

1

K =

+

2

K =

Hamiltonians Hamiltonians

f the
f the

Generators Generators

f the
f the

Ambient Group Ambient Group

Hamiltonians for Hamiltonians for ( ) A n

Each generator is the product of Each generator is the product of disjoint transpositions, i.e., disjoint transpositions, i.e.,

( ) g A n ∈

( ) ( )( ) ( )

1 1 2 2

, , , g K K K K K K

α β α β α β α β

=

(

) ( )( ) ( ) ( ) ( )

1 1 2 2 3 1

, , , g K K K K K K η η

− −

=

Choose a permutation so that

Choose a permutation so that

η

Hence, Hence,

1 1 1 1 2 n

g I σ σ η η η η σ

− −

        =        

1

1 1 σ   =    

, where , where

1 1 σ   =    

Also, let , and note that Also, let , and note that

For simplicity, we choose the branch . For simplicity, we choose the branch .

s =

( ) ( )

( ) ( ) ( ) ( )

1 1 2 2

2

n n

I σ σ σ σ π η η −

− × − × −

  ⊗ − ⊗ − =      

(

) ( )

1 1

ln

g

H i g η η η η η η

− − − −

= − = − Hamiltonians for Hamiltonians for ( ) A n

( ) ( ) ( ) ( )( ) ( )

1 1

ln 2 1 2 , i s s π σ σ σ σ σ = + = + − ∈

Log of a matrix

The Log of a Unitary Matrix The Log of a Unitary Matrix

Let U be an arbitrary finite rxr unitary matrix. Moreover, there exists a unitary matrix W which diagonalizes U, i.e., there exists a unitary matrix W such that

( ) ( )

1 2

1

, , ,

r

i i i

WUW e e e

θ θ θ θ θ − = ∆

= ∆ …

where are the eigenvalues of U. Then eigenvalues of U all lie on the unit circle in the complex plane.

1 2

, , ,

r

i i i

e e e

θ θ θ θ θ

…

Since , where is an arbitrary integer, we have

The Log of a Unitary Matrix The Log of a Unitary Matrix

( ) ( )

1 2

1

ln ln( ),ln( ), ,ln( )

r

i i i

U W e e e W

θ θ θ θ θ −

= ∆ = ∆ …

Then

( ) ( )

ln 2

j

i j j

e i in

θ

θ π θ π = + = +

j

n ∈

( ) ( ) ( ) ( )

1 1 1 2 2

ln 2 , 2 , , 2

r r

U iW n n n W θ π θ π θ π θ π

−

= ∆ = ∆ + + + …

where

1 2

, , ,

r

n n n ∈ …

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14

The Log of a Unitary Matrix The Log of a Unitary Matrix

( ) ( )

/ !

A m m

e A m

∞ =

= ∑

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1 1 1

ln , ,ln ln ln , ,ln 1 ln ln 1 2 2 1 1

, , , , , ,

r r r r r r

W i i W U i i i i i in i in i i

e e W e W W e e W W e e W W e e W U

θ θ θ θ θ θ θ θ θ θ θ θ θ π θ π θ π θ θ θ θ

− ∆

∆ − − + + + + − −

= = = ∆ = ∆ = ∆ = ∆ = ∆ = ∆ =

… …

… … …

Since , we have

Back

Hamiltonians for Hamiltonians for ( ) A n

Using the Using the Hamiltonian Hamiltonian for the for the Reidemeister Reidemeister 2 move 2 move

( ) ( )

2,1

↔

g =

cos 2 t π      

sin 2

t i π   −    

2

i t

e

π      

and the

and the initial state initial state we have that the we have that the solution to Schroedinger’s solution to Schroedinger’s equation equation for time is for time is

t

Observables Observables

which are which are

Quantum Knot Quantum Knot Invariants Invariants

Question Question: : Which knot invariants Which knot invariants are physically implementable ??? are physically implementable ???

Observable Q. Knot Invariants Observable Q. Knot Invariants

Question Question. What do we mean by a What do we mean by a physically observable knot invariant ? physically observable knot invariant ? Let be a quantum knot system. Let be a quantum knot system. Then a quantum observable is a Hermitian Then a quantum observable is a Hermitian

perator on the Hilbert space .
perator on the Hilbert space .

( ) ( )

( ),

( )

n A n

K Ω

( ) n

K Observable Q. Knot Invariants Observable Q. Knot Invariants

Question Question. But which observables are But which observables are actually knot invariants ? actually knot invariants ?

Ω

Def

Def. An observable is an

An observable is an invariant invariant of

f

quantum quantum knots knots provided for provided for all all

1

U U − Ω = Ω = Ω ( ) U A n ∈ Ω

( ) n

W = ⊕ K

be a decomposition of the representation

be a decomposition of the representation

( ) ( )

( )

n n

A n × → × → K K K K Observable Q. Knot Invariants Observable Q. Knot Invariants

Question Question. But how do we find quantum knot But how do we find quantum knot invariant observables ? invariant observables ? Theorem

Theorem. Let be a quantum

Let be a quantum knot system, and let knot system, and let

( ) ( )

( ),

( )

n A n

K

Then, for each , the projection operator Then, for each , the projection operator for the subspace is a quantum knot for the subspace is a quantum knot

bservable.
bservable.

P

W

into irreducible representations . into irreducible representations .

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15

Theorem

Theorem. Let be a quantum

Let be a quantum knot system, and let be an observable knot system, and let be an observable

n .
n .

( ) ( )

( ),

( )

n A n

K Ω

( ) n

K Observable Q. Knot Invariants Observable Q. Knot Invariants

Let be the stabilizer Let be the stabilizer subgroup for , i.e., subgroup for , i.e.,

Ω

( ) ( )

St Ω

( ) ( ) {

}

1

( ) : St U A n U U − Ω = Ω = ∈ Ω = Ω

Then the observable Then the observable

( ) ( )

1 ( )/ U A n St

U U −

∈ Ω ∈ Ω

Ω

∑

is a quantum knot invariant, where the is a quantum knot invariant, where the above sum is over a complete set of coset above sum is over a complete set of coset representatives of in . representatives of in .

( ) ( )

St Ω ( ) A n Observable Q. Knot Invariants Observable Q. Knot Invariants

Orbit Projectors: For each mosaic

knot K, the observable

( ) ' ( )

' '

A n K K A n K

P K K

∈

= ∑

Let be a knot invariant. Then

: I → K

(

) ( )

K

I I K K K

∈

= ∑ K(n) is an observable which is a quantum knot n-invariant. is a quantum knot n-invariant.

Observable Q. Knot Invariants Observable Q. Knot Invariants

+ Ω = Ω =

The following is an example of a quantum The following is an example of a quantum knot invariant observable: knot invariant observable: Quantum Quantum

≠

A Quantum Knot Invariant Quantum Invariant as is usually defined

An Alternate Approach An Alternate Approach

Next Objective Next Objective

We would like to find an definition

f quantum knots that is more directly

related to the spatial configurations of knots in 3-space. equivalent Rationale: Mosaics are based on knot projections, and hence, only indirectly on the actual knot. Moreover, PI & Reidemeister moves are moves on knot projections, and hence, also only indirectly associated with the actual spacial configuration of knots.

SLIDE 16

16

Reidemeister Reidemeister Moves: Moves: R1 R1 R2 R2 R3 R3 R0 R0

Can we find an alternative Can we find an alternative approach to knot theory ? approach to knot theory ?

Can we find an alternate approach to knots that is much more “physics friendly” ???

Can we find an alternative Can we find an alternative approach to knot theory ? approach to knot theory ?

Before we ask this question, we need to find the answer to a more fundamental question: How does a dog wag its tail ? How does a dog wag its tail ?

How does a dog wag its tail? How does a dog wag its tail?

1988 - 2000 My best friend My best friend Tazi Tazi knew the answer. knew the answer. Tazi = Tasmanian Tiger

How How does does a dog wag its tail ? a dog wag its tail ?

She would wiggle her tail, just

as a a creature would squirm on a flat planar surface.

She would wag her tail in a

twisting corkscrew motion.

Her tail would also stretch or

contract when an impolite child would tug on it.

Key Intuitive Idea Key Intuitive Idea

A curve in 3-space has 3 local (i.e., infinitesimal) degrees of freedom. Can we take this idea and use it to create a useable well-defined set of moves which will replace the Reidemeister moves ? Wiggle A curvature Move Wag A torsion Move Tug A metric move

SLIDE 17

17

Can we find an alternate set of knot moves that is much more “physics friendly” ???

Reidemeister Moves

Mechanical Engineers already have Mechanical Engineers already have 2) A wiggle: 3) A wag: 1) A tug: 3-Bar Prismatic Linkage 4-Bar Linkage 3-Bar Swivel Linkage For Mechanical Engineers, there are two types of knot theory. Extensible Knot Theory, where moves that change the length of the knot are permitted. Inextensible Knot Theory, where moves that change the length of the knot are not permitted. Knot Theory According to Mechanical Engineers Knot Theory According to Mechanical Engineers A tug 3-Bar Prismatic Linkage Wiggles and wags can be replaced by sequences

f tugs.

Hence, for extensible knot theory, there is

nly one move, i.e.,

which is the same as Reidemeister’s original triangle move. Knot Theory According to Mechanical Engineers Knot Theory According to Mechanical Engineers For inextensible knot theory, there are two moves, i.e.,

A wiggle:
A wag:

4-Bar Linkage 3-Bar Swivel Linkage We would like to discuss both

Extensible (a.k.a., topological) knot

theory, and

Inextensible Knot Theory.

But because of time constraints, we will focus mainly on extensible knot theory.

SLIDE 18

18

The Tangle is a device that moves only by wagging. The Bendangle is a device that moves only by

wiggling. (Patent pending)

The Universal Bendangle is a device that moves

nly by wiggling and wagging. (Patent pending)

Quantum Knots Quantum Knots Lattices Lattices &

The Cubic Honeycomb The Cubic Honeycomb A Scaffolding for 3 A Scaffolding for 3-

Space

Space

For each non-negative integer , let denote the 3-D lattice of points

L
3

1 2 1 2 3

, , : , , 2 2 2 m m m m m m     = ∈ = ∈        

L
lying in Euclidean 3-space

3

This lattice determines a tiling of by

3

2

2 2

− − − − − −

× × × ×

cubes,

called the cubic honeycomb of (of order )

3

The Cubic Honeycomb (of order )

The Cubic Honeycomb (of order ) A Scaffolding for 3 A Scaffolding for 3-

Space

Space

SLIDE 19

19

The Cubic Honeycomb The Cubic Honeycomb

Vertices

a∈ L

Edges

E

Faces

F

We think of this honeycomb as a cell complex for consisting of: C

3
Cubes

B

All cells of positive dimension are open cells.

Open Open Open

Lattice Knots Lattice Knots

Definition. A lattice graph G (of order

) is a finite subset of edges (together with their endpoints) of the honeycomb C

Definition. A lattice Knot K (of order

) is a lattice 2-valent graph (of order ).

Let denote the set of all lattice knots
f order

.

( )

K
Lattice Knots

Lattice Knots

Lattice Trefoil Lattice Hopf Link

Necessary Necessary Infrastructure Infrastructure

Orientation of 3 Orientation of 3-

Space

Space

We define an orientation of by selecting a right handed frame

3

3

e

2

e

1

e at the origin O = (0,0,0) properly aligned with the edges of the honeycomb, and by parallel transporting it to each vertex a∈ L

We refer to this frame as the preferred

frame.

3

e

2

e

1

e

3

e

2

e

1

e

Orientation of 3 Orientation of 3-

Space

Space

The preferred frame at each lattice point.

3

e

2

e

1

e O

SLIDE 20

20

Color Coding Conventions Color Coding Conventions for Vertices & Edges for Vertices & Edges

“Hollow” Gray Not part of Lattice Knot Solid Red Part of the Lattice Knot Solid Gray Indeterminant, maybe part of Lattice Knot A vertex a of a cube B is called a preferred vertex of B if the first octant of the preferred frame at a contains the cube B. Since B is uniquely determined by its preferred vertex, we use the notation

( )( )

B B a =

The preferred edges and preferred faces of

are respectively the edges and faces of that have a as a vertex

( )( )

B a

( )( )

B a

( )( )

p

F a =

Preferred face perpendicular to

p

e

( )( ) p

E a =

Preferred edge parallel to

p

e

Every edge is a preferred edge of exactly one cube
Every face is the preferred face of exactly one cube

Hence, the following notation uniquely identifies each edge and face of the cell complex C

2

e

3

e

1

e

( ) 3 ( )

E a

( )

2 ( )

E a

( )

3 ( )

F a

( )

1 ( )

E a

( )

2 ( )

F a

( )

1 ( )

F a

Preferred Vertices, Edges, & Faces

Preferred Vertices, Edges, & Faces

( )( )

B a

a

2

e

3

e

1

e

Drawing Conventions Drawing Conventions

( )( )

B a

a

When drawn in isolation, each cube is drawn with edges parallel to the preferred frame, and with the preferred vertex in the back bottom left hand corner.

( )( )

B a

Drawing Conventions

Drawing Conventions

2( )

e a

1( )

e a

a

( ) ( )

( ) 2

F a

2( )

e a

3( )

e a

a

( ) ( )

( ) 1

F a

2( )

e a

a

( ) ( )

( ) 3

F a

1( )

e a

3

e

1

2

e

1

e

2 3

a

SLIDE 21

21

The Left and Right Permutations The Left and Right Permutations

Define the left and right permutations and as

{ } { }

: 1,2,3 1,2,3 1 2 2 3 3 1 →

{

} { }

: 1,2,3 1,2,3 1 3 2 1 3 2 →

p

p p

e e e = × = ×

p p p

e e e = × = ×

p p p

e e e = × = ×

Ergo, ( )

p

e a ( )

p

e a

( )

p

e a

points out of the page toward the reader

Preferred Vertex Invisible preferred frame

a

( ) ( )

( ) p

E a

( )

( )

( ) p

E a

( )

( )

( ) p

F a

When drawn in isolation, is always

drawn with preferred vertex in the upper left hand corner, and with pointing out

f the page.

( )( ) p

F a

( )

p

e a

Drawing Conventions Drawing Conventions

a

( )( ) p

F a

( )( )

p

F a

( )( )

p

E a

( )( )

p

E a

( )( )

p

E a

( )( )

p

F a

a

p

e

p

e

p

e

Vertex Translation Vertex Translation Let be a vertex in the lattice

a

L

:

2

p p

a a e

−

= + = +

:

2

p p

a a e

−

= − = −

3

:

3 2

p p

a a e

−

= + = + ⋅

5

2

:1 2 3 1 2 3

2 2 5 2 2 a a e e e

− − − − −

= + = + ⋅ − ⋅ +

So for example,

Lattice Knot Lattice Knot Moves Moves

Lattice knot moves Lattice knot moves

Definition

Definition. A lattice knot move µ (of
rder

) is a bijection

( )

( )

: µ →

K

K

The move µ is said to be local if there exists a closed cube in the lattice such that

( )( )

B a

( )

( )

( ) ( ) K B a K B a

id µ

− − − −

=

( )

K ∈

K

for all

SLIDE 22

22

Lattice Knot Moves Lattice Knot Moves

We will now define the lattice knot moves tug, wiggle, and wag.

Tugs Tugs

( ) ( )

( ) p

F a

(

) ( )

( ) 1

, , L a p

This is a local move on face

( ) ( )

( ) p

F a

(

) ( )

( )

, a p =

(

) ( ) ( )

( ) ( ) 1

, , , L a p a p =

Tugs

Tugs

( ) ( )

( ) p

F a

(

) ( )

( ) , ( )

a p K =

(

) ( ) ( ) ( )

if if

therwise

K K K K K − = − = − = − = ∪ ∩ ∪ ∩

means For each cube, 4 Tugs for each preferred face For each cube, 4 Tugs for each preferred face

( ) ( )

( ) p

F a

(

) ( )

( )

, a p

( )

( )

( ) p

F a

(

) ( )

( )

, a p

( )

( )

( ) p

F a

(

) ( )

( )

, a p

( )

( )

( ) p

F a

(

) ( )

( )

, a p

12 tugs for each cube

Tugs are Tugs are extensible extensible local moves local moves For each cube, 2 Wiggles for each preferred face For each cube, 2 Wiggles for each preferred face

( ) ( )

( ) p

F a

6 wiggles per cube

( ) ( )

( )

, a p

( )

( )

( ) p

F a

(

) ( )

( )

, a p

(

) ( )

2

, , L a p =

( ) ( )

2

, , L a p =

Wiggles are Wiggles are inextensible inextensible local moves local moves ( ) 1

( ) F a

(

) ( ) ( )

( ) ( ) 3

,1, ,1 L a a =

Wags

Wags

3

e

2

e

1

e

Hinge

a

Hinge Joint

a

Hinge Joint

( ) ( )

( ) 1

F a

2

e

3

e

a

Bottom Face

( ) ( )

( ) 1

F a

3

e

2

e

a

Back Face

Hinge

SLIDE 23

23

The Left and Right Permutations The Left and Right Permutations

Please recall that the left and right permutations and are defined as

{ } { }

: 1,2,3 1,2,3 1 2 2 3 3 1 →

{

} { }

: 1,2,3 1,2,3 1 3 2 1 3 2 →

p

p p

e e e = × = ×

p p p

e e e = × = ×

p p p

e e e = × = ×

( )( ) p

F a

( )( )

p

F a

( )( )

p

E a

( )( )

p

E a

( )( )

p

E a

( )( )

p

F a

a

p

e

p

e

p

e

Preferred edges & Faces

p

e

p

e

p

e

Preferred Vertex a Invisible preferred frame = Edge 0

⊥

= Edge 1

⊥

= Edge 2

⊥

= Edge 3

⊥ ( ) ( )

( ) p

F a

( )

1

( ) F a

(

) ( )

( )

,1 a

( )

1

( ) F a

(

) ( )

( )

,1 a

( )

1

( ) F a

(

) ( )

( )

,1 a

( )

1

( ) F a

(

) ( )

( )

,1 a

4 Wags for face 1

4 Wags for face 1 Wags Wags For each cube, there are 4 wags for each of its 3 preferred faces. Hence, there are 12 wags per cube. Wags are Wags are inextensible inextensible local moves local moves

An Example of a Wiggle Move An Example of a Wiggle Move K ( ) ( )(

) ( )

( ) :1,3

a K

1

e

2

e

( ) ( )

( ) :1,3

a

Wiggle

Face

( ) ( )(

) ( )

:1 3

F a

Vertex

:1

a

These are Conditional Moves a

SLIDE 24

24

The The Ambient Ambient Groups Groups

Tug, Wiggle, & Wag are Permutations Tug, Wiggle, & Wag are Permutations

For each , each of the above moves, Tug, Wiggle, & Wag, is a permutation (bijection) on the set

f all lattice knots of order .

≥

( )
K
In fact, each of the above local moves,

as a permutation, is the product of disjoint transpositions.

Definition. The ambient group

is the group generated by tugs, wiggles, and wags of order .

The Ambient Groups and The Ambient Groups and

Λ
Λ

Λ

Λ
Tugs, wiggles, and wags are a set

Tugs, wiggles, and wags are a set

f involutions that generate the
f involutions that generate the

above groups. above groups.

Definition. The inextensible ambient group

is the group generated only by wiggles and wags of order .

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

, , , , , , , a p a p a p a p a p a p a p = =

(

) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

, , , , , , , a p a p a p a p a p a p a p = =

(

) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

, , , , , , , a p a p a p a p a p a p a p = =

(

) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

, , , , , , , a p a p a p a p a p a p a p = =

What Is the Ambient Group ?

What Is the Ambient Group ?

We create a representation of the Ambient Group onto a group of Conditional OP Autohomeomorphisms of .

3

What is the ambient group ???

What is the ambient group ???

Observation: Wiggle, wag, and tug are symbolic conditional moves, For example, the tug as are the Reidemeister moves.

( ) ( )

( )

, a p =

(

) ( ) ( ) ( )

if if

therwise

K K K K K − = − = − = − = ∪ ∩ ∪ ∩

Conditions

SLIDE 25

25

What is the ambient group ??? What is the ambient group ???

Observation: Each is a symbolic representation of an authentic conditional move

3

Moreover, each involved (OP) auto-

homeomorphism is local, i.e., there exists a 3-ball D such that

3 3

: h →

3

3 3

:

D

h id D D

−

= − = − → −

, i.e., a conditional orientation

preserving (OP) auto-homeomorphism of . Let be a family of knots in .

What is the ambient group ??? What is the ambient group ???

Let the group of local OP autohomeomorphisms of .

( ) ( )

3 OP

LAH

3
F

For example:

( )

K

The family of lattice knots S The family of finitely piecewise smooth (FPWS) knots in .

3

3
What is the ambient group ???

What is the ambient group ???

Def. A local authentic conditional (LAC) move on a family of knots is a map F

( ) ( ) ( ) ( )

3 3 3

: :

OP K

LAH K Φ → Φ → Φ → Φ →

F

such that

( ) ( )

K K

K Φ ∈ Φ ∈ ∀ ∈ F F F F Let be the space of all LAC moves for the family .

( ) ( )

3 OP

LAH

F

F

What is the ambient group ??? What is the ambient group ???

Let be the space of all LAC moves for the family .

( ) ( )

3 OP

LA H

F

F Define a multiplication ‘ ‘ as follows: i

( ) ( ) ( ) ( ) ( )

( ) ( )

3 3 3

', '

OP OP OP

LAH LAH LAH × → × → Φ Φ Φ Φ Φ Φ

i

F F F F F

as

( ) ( )

( )

' '

K K

K K Φ

Φ Φ Φ Φ = Φ Φ i

where ‘ ‘ denotes the composition of

functions.

What is the ambient group ???

What is the ambient group ???

Proposition. is a monoid.

( ) ( )

( )

3

,

O P

LA H

i

F

In the paper “Quantum Knots and Lattices,” we construct a faithful representation

( ) ( )

3

:

OP

LAH Γ Λ Γ Λ →

F

into a subgroup of the moniod by mapping each generator wiggle, wag, and tug onto a local conditional OP auto- homeomorphism of .

( ) ( )

( )

3

,

OP

LA H

i

F

3

Refinement

Refinement

SLIDE 26

26

The refinement injection The refinement injection

( ) ( 1)

:

+

→ K K Q

Def.

We define the refinement injection from lattice knots of order to lattice knots of order as

1

+

{

} { }

( )

( ) ( 1) 3 ( ) ( ) : 1 ( )

: ( ), ( )

p

p p p a p E a

K E a E a

+ ∈ = ∈ =

→ K K Q

∪ ∪ ∪

L

( ) ( ) (

) ( )

:1 3

F a

1

e

2

e

2

[ ] [ ] ( ) ( )

*

, F x a f δ δ

[ ] [ ] ( ) ( )

*

, F x a f δ δ

The refinement injection The refinement injection

An example:

Q

( )

K

( 1) +

K

Conjectured Refinement Conjectured Refinement Monomorphism Monomorphism

We conjecture the existence of a refinement monomorphism

1

:

+

Λ → Λ → Λ Q

which preserves the action

( ) ( )

, g K gK Λ × Λ × → K K

i.e., with the property

( ) ( ) ( ) ( )

g K gK = Q Q Q Q Q In fact, we have a construction which we believe is such a monomorphism.

( ) ( )

? ( )

,1 a = Q

The Refinement Morphism ???

The Refinement Morphism ???

1

e

2

e

( ) ( )( )

( )

1

F a

(

) ( ) ( ) ( ) ( )

( ) ( )

( 1) :23 ( 1) :2 ( 1) :3 ( 1)

,1 ,1 ,1 ,1 a a a a

+ + + + + + +

1

:

+

Λ → Λ → Λ Q

^

^ ^

1

^ a b b ab

−

=

Knot Type Knot Type

SLIDE 27

27

Lattice Knot Type Lattice Knot Type

Two lattice knots and in are of the same

type, written

1

K

2

K

( )

K

1

2

~ K K

provided there is an element such that

g ∈ Λ ∈ Λ

1 2

gK K = They are of the same knot type, written

1 2

~ K K provided there is a non-negative integer such that m

1 2

~

m m m

K K

+

Q Q Q Q

Inextensible Lattice Knot Type

Inextensible Lattice Knot Type

Two lattice knots and in are of the same inextensible

type, written

1

K

2

K

( )

K

1

2

K K ≈

provided there is an element such that
g ∈ Λ

∈ Λ

1 2

gK K = They are of the same inextensible knot type, written

1 2

K K ≈ provided there is a non-negative integer such that m

1 2 m m m

K K

+

≈ Q Q Q Q

n-
Bounded Lattices,

Bounded Lattices, Lattice knots, and Lattice knots, and Ambient Groups Ambient Groups

In preparation for creating a definition of physically emplementable quantum knot systems, we need to work with finite mathematical objects. n-

Bounded Lattices

Bounded Lattices

{ }

,

:

n

a a n

∞

= ∈ = ∈ ≤

L

L L L Let and be be non-negative integers. We define the n-bounded lattice of order as

n
where

( ) ( )

max j

j

a a

∞ =

We also have

,n

C

The corresponding cell complex

, j n

C

The corresponding j-skeleton

n-

Bounded Lattice Knots &

Bounded Lattice Knots & Ambient Groups Ambient Groups

( , ) ( ) , n n

= ∩ = ∩ K K

L

set of n-bounded lattice knots of order

,

n

Λ = Λ = Λ

C
,

,

n

Λ = Λ = Λ

C

Ambient group of order (

) ( )

,n

Inextensible Ambient group
f order (

) ( )

,n

SLIDE 28

28

n-

Bounded Lattice Knots &

Bounded Lattice Knots & Ambient Groups Ambient Groups

We also have the injection

( , ) ( , 1)

:

n n

ι

+

→ K K

( , )

( , 1)

:

n n

ι

+

Λ → Λ → Λ

( , )

( , 1)

:

n n

ι

+

Λ → Λ → Λ

and the monomorphisms

( ) ( ) ( ) ( )

( ,1) ( ,2) ( , ) ,1 ,2 ,

, , ,

n n

Λ → Λ → Λ → → Λ → K K K

We thus have a nested sequence of lattice

knot systems and

n-

Bounded Lattice Knot Type

Bounded Lattice Knot Type

Two lattice knots and in are said to be of the same lattice knot

type, written

1

K

2

K

( , ) n

K

,n

g ∈ Λ ∈ Λ

( ) ( )

,n

1

2

~

n

K K

provided there is an element such that

1 2

gK K = They are of the same lattice knot type, written

1 2

~ K K provided there are non-negative integers and such that '

'

n

' ' ' 1 2 '

~

n n n n

K K ι ι ι ι

+ +

Q Q Q Q

n-
Bounded Lattice Knot Type

Bounded Lattice Knot Type

In like manner for the inextensible ambient group , we can define

1 2 n

K K ≈

1

2

K K ≈ ,n Λ and

Quantum Knots Quantum Knots & Quantum Knot Systems Quantum Knot Systems

The Hilbert Space The Hilbert Space

f Quantum Knots
f Quantum Knots

( , ) n

K

For Q.M. systems, we need an underlying Hilbert space. So we define: Recall that is the set of all lattice knots of order .

( , ) n

K

( , ) n

The

The Hilbert Hilbert space space

f
f quantum

quantum knots knots is the Hilbert space with the set of lattice knots of order as its

rthonormal basis, i.e., with orthonormal

( , ) n

K

( , ) n

K

( , ) n

{

}

( , )

:

n

K K ∈

K

Quantum Knots Quantum Knots

It’s time to remodel the bounded lattice by painting all its edges.

,n

L

Two available cans of paint

“Solid” Red “Hollow” Gray An Edge A Non A Non-

Edge

Edge Set of all 2- colorings of edges of

,n

L

Set of all

lattice graphs in

( , ) n

K

,n

L

Identification

SLIDE 29

29

Quantum Knots Quantum Knots = E

2-D Hilbert space with orthonormal basis

1 = 0 =

Non-Edge Existent Edge

“Hollow” Gray “Solid” Red

Edge Coloring Space

E

Hilbert Space of Lattice Graphs in

( , ) n

G

,n

L

(

) ( )

( ) ,

( , )

n

n E Edges ∈

=

⊗

L

G E G E

Quantum Knots

Quantum Knots

Hilbert Space of Lattice Graphs in

( , ) n

G

,n

L

(

) ( )

( ) ,

( , )

n

n E Edges ∈

=

⊗

L

G E G E

(

) ( )

( ) ( ) ( ) ( )

{ } { }

( ) ,

( ) ( ) ,

, | : ,

n

n E Edges

E c E c Edges

∈

    →      

⊗

L

Gray

Red

{ }

,

lattice graph i | n

n

G G L

Orthonormal basis is: which is identified with

Quantum Knots Quantum Knots

Hilbert Space of Lattice Graphs in

( , ) n

G

,n

L

(

) ( )

( ) ,

( , )

n

n E Edges ∈

=

⊗

L

G E G E

{

}

,

lattice graph i | n

n

G G L

Orthonormal basis is:

Quantum Knots Quantum Knots

Hilbert Space of Lattice Graphs in

( , ) n

G

,n

L

(

) ( )

( ) ,

( , )

n

n E Edges ∈

=

⊗

L

G E G E

{

}

,

lattice graph i | n

n

G G L

Orthonormal basis is:

Hilbert Space of quantum knots

( , ) n

K

( , ) n =

K

Sub-Hilbert space of with

rthonormal basis

( , ) n

G

{ }

( , )

|

k

K K ∈

K

+

K = 2 An Example of a Quantum Knot An Example of a Quantum Knot

Since each element is a permutation, Since each element is a permutation, it is a linear transformation that simply it is a linear transformation that simply permutes basis elements. permutes basis elements.

,n

g∈Λ ∈Λ The Ambient Group as a The Ambient Group as a Unitary Unitary Group Group

,n

Λ

We We identify identify each element with the each element with the linear transformation defined by linear transformation defined by

,n

g∈ Λ ∈ Λ

( , ) ( , ) n n

K gK → K K K K

Hence, under this identification, the

Hence, under this identification, the ambient ambient group group becomes a becomes a discrete group discrete group of

f

unitary transfs unitary transfs on the Hilbert space .

n the Hilbert space .

( , ) n

K

,n

Λ

SLIDE 30

30

An Example of the Group Action An Example of the Group Action K =

,n

Λ

+

2 ( ) ( )( ) ( )

( ) :1,3

a K =

1

e

2

e

( ) ( )

( ) :1,3

a

Wiggle

+

2

Face

( ) ( )(

) ( )

:1 3

F a

Vertex

:1

a The Quantum Knot System The Quantum Knot System (

) ( )

( , ) ,

,

n n

Λ K

(

) ( ) ( ) ( )

( ,1) ( , ) ( , 1) ,1 , , 1

, , ,

n n n n ι ι ι ι ι ι + +

Λ → Λ → → Λ → Λ → K K K K K

Physically

Physically Implementable Implementable

Def Def. A A quantum quantum knot knot system system is a is a quantum system having as its state space, quantum system having as its state space, and having the Ambient group as its set and having the Ambient group as its set

f accessible unitary transformations.
f accessible unitary transformations.

( , ) n

K

,n

Λ

( ) ( )

( , ) ,

,

n n

Λ K

Physically

Physically Implementable Implementable Physically Physically Implementable Implementable

The states of quantum system are The states of quantum system are quantum quantum knots

knots. The elements of the ambient

. The elements of the ambient group are group are quantum quantum moves moves.

( ) A n

( ) ( )

( , ) ,

,

n n

Λ K

The Quantum Knot System

The Quantum Knot System (

) ( )

( , ) ,

,

n n

Λ K

(

) ( ) ( ) ( )

( ,1) ( , ) ( , 1) , , , 1

, , ,

n n n n n ι ι ι ι ι ι + +

Λ → Λ → → Λ → Λ → K K K K K

Physically

Physically Implementable Implementable Physically Physically Implementable Implementable Physically Physically Implementable Implementable

Choosing integers and n is analogous to choosing respectively the thickness and the length of the rope. The parameters (wiggle, wag, & tug) of the ambient group are the “knobs” one turns to spacially manipulate the quantum knot.

,n

Λ

The smaller the thickness and the longer the rope, the more knots that can be tied.

Quantum Knot Type

Quantum Knot Type

Def Def. Two quantum knots and are Two quantum knots and are

f the
f the same

same knot knot

type

type, written , written

1

K

2

K

1 2 , n

K K

∼

provided there is an element provided there is an element s.t s.t. .

,n

g∈ Λ ∈ Λ

1 2

g K K =

They are of the They are of the same same knot knot type type, written , written

1 2 ,

K K ∼

' ' ' ' ' 1 2 ' n n n n

K K ι ι ι ι

+ +

∼

Q Q Q Q

provided there are integer provided there are integer such that such that

', ' n ≥

(

) ( )

,n

K =

+

2 ( ) ( )( ) ( )

( ) :1,3

a K =

1

e

2

e

( ) ( )

( ) :1,3

a

Wiggle

+

2 Two Quantum Knots of the Same Knot Type Two Quantum Knots of the Same Knot Type

Two Quantum Knots NOT of the Same Knot Type Two Quantum Knots NOT of the Same Knot Type

1

K =

+

2

K =

SLIDE 31

31

Hamiltonians Hamiltonians

f the
f the

Generators Generators

f the
f the

Ambient Group Ambient Group

Hamiltonians for Hamiltonians for ( ) A n

Each generator is the product of Each generator is the product of disjoint transpositions, i.e., disjoint transpositions, i.e.,

,n

g∈ Λ ∈ Λ

( ) ( )( ) ( )

1 1 2 2

, , ,

r r

g K K K K K K

α β α β α β α β α β

=

(

) ( )( ) ( ) ( ) ( )

1 1 2 3 4 1

, , ,

r r

g K K K K K K η η

− −

=

Choose a permutation so that

Choose a permutation so that

η

Hence, Hence,

( ) ( )

1 1 1 1 , 2 d n r

g I σ σ η η η η σ

− −

        =        

1

1 1 σ   =    

, where , where ( ) ( )

( ) ( )

( , )

, dim

n

d n = K

1

1 σ   =    

Also, let , and note that Also, let , and note that

For simplicity, we always choose the branch . For simplicity, we always choose the branch .

s =

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 , 2 , 2

2

r d n r d n r

I σ σ σ σ π η η −

− × − × −

  ⊗ − ⊗ − =      

(

) ( )

1 1

ln

g

H i g η η η η η η

− − − −

= − = − Hamiltonians for Hamiltonians for ( ) A n

( ) ( ) ( ) ( )( ) ( )

1 1

ln 2 1 2 , i s s π σ σ σ σ σ = + = + − ∈

Matrix log def

The Log of a Unitary Matrix The Log of a Unitary Matrix

Let U be an arbitrary finite rxr unitary matrix. Moreover, there exists a unitary matrix W which diagonalizes U, i.e., there exists a unitary matrix W such that

( ) ( )

1 2

1

, , ,

r

i i i

WUW e e e

θ θ θ θ θ − = ∆

= ∆ …

where are the eigenvalues of U. Then eigenvalues of U all lie on the unit circle in the complex plane.

1 2

, , ,

r

i i i

e e e

θ θ θ θ θ

…

Since , where is an arbitrary integer, we have

The Log of a Unitary Matrix The Log of a Unitary Matrix

( ) ( )

1 2

1

ln ln( ),ln( ), ,ln( )

r

i i i

U W e e e W

θ θ θ θ θ −

= ∆ = ∆ …

Then

( ) ( )

ln 2

j

i j j

e i in

θ

θ π θ π = + = +

j

n ∈

( ) ( ) ( ) ( )

1 1 1 2 2

ln 2 , 2 , , 2

r r

U iW n n n W θ π θ π θ π θ π

−

= ∆ = ∆ + + + …

where

1 2

, , ,

r

n n n ∈ …

The Log of a Unitary Matrix

The Log of a Unitary Matrix

( ) ( )

/ !

A m m

e A m

∞ =

= ∑

( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1 1 1

ln , ,ln ln ln , ,ln 1 ln ln 1 2 2 1 1

, , , , , ,

r r r r r r

W i i W U i i i i i in i in i i

e e W e W W e e W W e e W W e e W U

θ θ θ θ θ θ θ θ θ θ θ θ θ π θ π θ π θ θ θ θ

− ∆

∆ − − + + + + − −

= = = ∆ = ∆ = ∆ = ∆ = ∆ = ∆ =

… …

… … …

Since , we have

Back

SLIDE 32

32

Hamiltonians for Hamiltonians for

,n

Λ

Using the Using the Hamiltonian Hamiltonian for the for the wiggle move wiggle move

cos 2 t π      

sin 2

t i π   −    

2

i t

e

π      

and the

and the initial state initial state we have that the we have that the solution to Schroedinger’s solution to Schroedinger’s equation equation for time is for time is

t ( ) ( )

( ) :1,3

a =

1

e

2

e ( ) ( )(

) ( )

:1 3

F a

Observables

Observables

which are which are

Quantum Knot Quantum Knot Invariants Invariants

Observable Q. Knot Invariants Observable Q. Knot Invariants

Question Question. What do we mean by a What do we mean by a physically observable knot invariant ? physically observable knot invariant ? Let be a quantum knot system. Let be a quantum knot system. Then a quantum observable is a Hermitian Then a quantum observable is a Hermitian

perator on the Hilbert space .
perator on the Hilbert space .

( ) ( )

( , ) ,

,

n n

Λ K

Ω

( , ) n

K Observable Q. Knot Invariants Observable Q. Knot Invariants

Question Question. But which observables are But which observables are actually knot invariants ? actually knot invariants ?

Ω

Def

Def. An observable is an

An observable is an invariant invariant of

f

quantum quantum knots knots provided for provided for all all

1

U U − Ω = Ω = Ω

,n

U ∈ Λ ∈ Λ Ω

( , ) n r r

W = ⊕ K

be a decomposition of the representation

( , ) ( , ) , n n n

Λ × Λ × → K K K K

Observable Q. Knot Invariants

Observable Q. Knot Invariants

Question. But how do we find quantum knot invariant observables ?

( ) ( )

( , ) ,

,

n n

Λ K

Theorem. Let be a quantum

knot system, and let Then, for each , the projection operator for the subspace is a quantum knot

bservable.

r

P

r

W

into irreducible representations . Let be the stabilizer subgroup for , i.e.,

Ω

( ) ( )

St Ω

Theorem. Let be a quantum

knot system, and let be an observable

n .

( ) ( )

( , ) ,

,

n n

Λ K

Ω

( , ) n

K Observable Q. Knot Invariants Observable Q. Knot Invariants

( ) ( ) {

}

1

( ) : St U A n U U − Ω = Ω = ∈ Ω = Ω

Then the observable

( ) ( )

,

1 /

n

U St

U U −

∈Λ ∈Λ Ω

Ω

∑

is a quantum knot invariant, where the

above sum is over a complete set of coset representatives of in .

( ) ( )

St Ω

,n

Λ

SLIDE 33

33

Observable Q. Knot Invariants Observable Q. Knot Invariants

In , the following is an example of an inextensible quantum knot invariant

bservable:

( ) ( ) ( ) ( )

3 3 ( ) ( ) ( ) : ( ) : 1 1 p p p p p p p p

F a F a F a F a

= = = =

Ω = Ω = ∂ ∂ + ∂ ∂

∑ ∑

where denotes the boundary of the

face .

( ) ( )

( ) p

F a ∂

( )

( )

( ) p

F a

( , )

n

K

Future Directions Future Directions & Open Questions Open Questions

Future Directions & Open Questions Future Directions & Open Questions

What is the structure of the ambient groups

What is the structure of the ambient groups , , , , and their direct limits ? , , , , and their direct limits ?

Λ

Can one find a presentation of these groups ? Can one find a presentation of these groups ? Are they Are they Coxeter Coxeter groups? groups?

Λ

,n

Λ ,n Λ

Exactly how are the lattice and the mosaic

Exactly how are the lattice and the mosaic ambient groups related to one another ? ambient groups related to one another ? Can one find a Can one find a presentation of the direct limits of these presentation of the direct limits of these groups ? groups ?

Future Directions & Open Questions Future Directions & Open Questions

Unlike classical knots, quantum knots can

exhibit the non-classical behavior of quantum superposition and quantum entanglement. Are quantum and topological entanglement related to one another ? If so, how ?

Future Directions & Open Questions Future Directions & Open Questions

For each lattice knot

For each lattice knot K, let denote , let denote its Jones polynomial. its Jones polynomial. We define the We define the Jones Jones

bservable
bservable as

as How is the Jones observable related to the Aharonov, Jones, Landau algorithm ?

( ) ( )

K

V t

( )

( ) ( ) ( )

( ) n K K

V t V t K K

∈

= ∑ K(n) Can it be used to create a different quantum algorithm for the Jones polynomial ?

Future Directions & Open Questions Future Directions & Open Questions

What is gained by extending the definition
f quantum knot observables to POVMs ?
What is gained by extending the definition
f quantum knot observables to mixed

ensembles ?

SLIDE 34

34

Future Directions & Open Questions Future Directions & Open Questions

Def. We define the lattice number of a knot

K as the smallest integer n for which K is representable as a lattice knot of order How is the lattice number related to the mosaic number of a knot? How does one compute the lattice number of a knot? How does one find a quantum

bservable for the lattice number?

( ) ( )

0,n =

Future Directions & Open Questions

Future Directions & Open Questions

Quantum Knot Tomography: Given many copies of the same quantum knot, find the most efficient set of measurements that will determine the quantum knot to a chosen tolerance . ε > Quantum Braids: Use lattices to define quantum braids. How are such quantum braids related to the work of Freedman, Kitaev, et al on anyons and topological quantum computing?

Future Directions & Open Questions Future Directions & Open Questions

Can quantum knot systems be used to model

and predict the behavior of

Quantum vortices in supercooled helium 2 ?
Fractional charge quantification that is

manifest in the fractional quantum Hall effect

Quantum vortices in the Bose-Einstein

Condensate This question can be answered in the positive by finding a Hamiltonian H for a quantum knot that predicts the behavior of a quantum vortex. Tug, Wiggle, & Wag are “physics friendly” Reason: From these moves, we can create by taking the limit as → ∞ → ∞

Variational derivatives w.r.t. moves, e.g.,
Infinitesimal moves, e.g.
Move differential forms, e.g.,
Multiplicative integrals of diff. forms,e.g.,

[ ] [ ] ( ) ( )

*

, F x a f δ δ

( ) ( )

1 1

* x x

x

∂ ∂ ∂ ∂ ⊗ ∂ ∂ ∂ ∂

( ) ( )

1 2

* dx dx

x

( ) ( )

1 2

, ,0

dx dx

x x

1 0..1

x =

2 0..1

x =

[ ] [ ] ( ) ( )

*

, F x a f δ δ

[ ] [ ] ( ) ( )

*

, F x a f δ δ Variational Derivatives w.r.t. moves

[ ] [ ] ( ) ( )

*

, F x a f δ δ

{ } {

} { }

( ) ( ) ( )

( ) ( ) ( ) B a a E a F a ∪ ∪ ∪ ∪ ∪

( )

PV a

Half Closed Cube Preferred Vertex

The Preferred Vertex (PV) Map The Preferred Vertex (PV) Map

−    

Since the half closed cubes form a partition of 3-space , we have the preferred vertex map

3

( )

3

: PV →

3

1 2 1 2 3

, , : , , 2 2 2 m m m m m m     = ∈ = ∈        

L

SLIDE 35

35

A finitely piecewise smooth (FPS) knot is a knot x in 3-space that consists of finitely many piecewise smooth ( ) segments with no two consecutive segments meeting in a tangential cusp. C ∞

3

Variational Derivatives

Let the family of FPS knots Let be a real valued functional

n

FPS =

F :

FPS

F → F

FPS

F

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) *

, , , , , a f a f a f a f a f =

(

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) *

, , , , , a f a f a f a f a f =

(

) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) *

, , , a f a f a f =

Total Tug, Total Wiggle, & Total Wag

Total Tug: Total Wiggle: Total Wag:

[ ] [ ] ( ) ( ) ( ) ( ) ( ) [ ] [ ]

( ) ( ) * 2 *

, 2 ,

lim

F a f PV x F x F x a f δ δ

− →∞ →∞

  −     =

[ ]

[ ] ( ) ( ) ( ) ( ) ( ) [ ] [ ]

( ) ( ) * 2 *

, 2 ,

lim

F a f PV x F x F x a f δ δ

− →∞ →∞

  −     =

Variational Derivatives w.r.t. moves

[ ] [ ] ( ) ( ) ( ) ( ) ( ) [ ] [ ]

( ) ( ) * 2 *

, 2 ,

lim

F a f PV x F x F x a f δ δ

− →∞ →∞

  −     =

Conjecture:

A functional is a knot invariant if all its variational derivatives exist and are zero. :

FPS

F → F

Variational Derivatives w.r.t. moves

UMBC UMBC Quantum Knots Quantum Knots Research Lab Research Lab

We at UMBC are very proud of our We at UMBC are very proud of our new state of the art Quantum Knots new state of the art Quantum Knots Research Laboratory. Research Laboratory. We have just purchased some of the latest and most advanced equipment in quantum knots research !!!

SLIDE 36

36

Weird !!! Weird !!!