Quantum Knots ??? An Intuitive Overview of the Theory of...

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Quantum Quantum Knots ???Knots ??? An Intuitive OverviewAn Intuitive Overview

of the Theory ofof the Theory ofQuantum KnotsQuantum Knots

Samuel LomonacoSamuel LomonacoUniversity of Maryland Baltimore County (UMBC)University of Maryland Baltimore County (UMBC)

Email: [email protected]: [email protected]: www.csee.umbc.edu/~lomonaco: www.csee.umbc.edu/~lomonaco

LL--OO--OO--PP

This is work in collaboration withLouis Kauffman

• Lecture II: Quantum Knots and Mosaics,

• Lecture III: Quantum Knots & Lattices,

• Lecture I: A Rosetta Stone for QuantumComputing

• Lecture IV: Intuitive Overview of the Theoryof 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 quantto appear soon on quant--phph

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. 2Processing, vol. 7, Nos. 2--3, (2008), 853, (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, NorthNorth--Holland, (1975), 217Holland, (1975), 217--2333.2333.

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

KitaevKitaev, Alexei Yu,, Alexei Yu, FaultFault--tolerant quantum computation tolerant quantum computation by by anyonsanyons,, http://arxiv.org/abs/quanthttp://arxiv.org/abs/quant--ph/9707021ph/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. on Quantum Information & Computation II (ed. by on Quantum Information & Computation II (ed. by DonkorDonkor, , PirichPirich, & Brandt), (2004), 5436, & Brandt), (2004), 5436--30, 26830, 268--284. 284. http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0403228ph/0403228 Throughout this talk:

“Knot” means either a knot or a link

PreamblePreamble

Thinking Outside the BoxThinking Outside the Box

Knot TheoryKnot Theory

Quantum MechanicsQuantum Mechanics

is a tool for exploringis a tool for exploring

ObjectivesObjectives

•• 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 GameRules of the Game

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

•• Physically meaningful, i.e., physicallyPhysically meaningful, i.e., physicallyimplementable, andimplementable, and

•• Simple enough to be workable and Simple enough to be workable and useable.useable.

3

AspirationsAspirations

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 occur in quantum physics such asoccur in quantum physics such as

•• In supercooled helium IIIn supercooled helium II

•• In the BoseIn the Bose--Einstein CondensateEinstein Condensate

•• In the Electron fluid found within the In the Electron fluid found within the fractional quantum Hall effectfractional quantum Hall effect

KnotTheory

QuantumMechanics

GroupRepresentation

Theory=

FormalRewritingSystem

=

FormalRewritingSystem

= GroupRepresentation

ThemesThemes

OverviewOverview•• PreamblePreamble

•• Mosaic KnotsMosaic Knots•• Quantum Mechanics: Whirlwind TourQuantum Mechanics: Whirlwind Tour

•• Preamble to Lattice KnotsPreamble to Lattice Knots•• Lattice KnotsLattice Knots

•• Quantum Knots & Quantum Knot SystemsQuantum Knots & Quantum Knot Systems

•• Q. Knots & Q. Knot SystemsQ. Knots & Q. Knot Systems

•• Future Directions & Open QuestionsFuture Directions & Open Questions

via Mosaicsvia Mosaics

via Latticesvia Lattices

Mosaic KnotsMosaic KnotsTransforming Knot Theory into Transforming Knot Theory into

a formal Rewriting Systema 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. 2Processing, vol. 7, Nos. 2--3, (2008), 853, (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

MosaicMosaic TilesTilesLet denote the following set of Let denote the following set of 1111symbols, called symbols, called mosaicmosaic ((unorientedunoriented) ) tilestiles::

( )uT

Please note that, up to rotation, there are Please note that, up to rotation, there are exactly exactly 55 tilestiles

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Mosaic KnotsMosaic Knots

A 4-mosaic trefoil

Figure Eight KnotFigure Eight Knot 55--MosaicMosaic

Hopf LinkHopf Link 44--MosaicMosaic

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

SubSub--Mosaic MovesMosaic Moves

A Cut & Paste MoveA Cut & Paste Move

1P←→

5

1P←→


1P←→


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

Planar Planar IsotopyIsotopy MovesMovesasas

SubSub--Mosaic MovesMosaic Moves

11 Planar Isotopy (PI) Moves on Mosaics11 Planar Isotopy (PI) Moves on Mosaics

3P←→

4P←→ 5P←→

7P←→

8P←→ 9P←→

2P←→

1P←→

6P←→

10P←→ 11P←→

Planar Isotopy (PI) Moves on MosaicsPlanar 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 subsub--mosaics by mosaics by 00, , 9090, , 180180, or , or 270270 degrees.degrees.

2 2×

For example,For example,

represents each of the followingrepresents each of the following 44 moves:moves:

1P←→

1P←→ 1P←→

1P←→ 1P←→

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

Each of the PI Each of the PI 22--submosaic moves represents submosaic moves represents any one of the any one of the (n(n--2+1)2+1)22 possible moves on an possible moves on an nn--mosaicmosaic ReidemeisterReidemeister MovesMoves

asasSubSub--Mosaic MovesMosaic Moves

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

2R←→

1R←→ '1R←→

'2R←→

''2R←→

'''2R←→

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

3R←→ '3R←→

'''3R←→

''3R←→

( )3ivR←→

( )3vR←→

Each Each PI move PI move and each and each R moveR move is a is a permutationpermutation of the set of all of the set of all knot knot nn--mosaics mosaics K(n)

PI & R Moves on MosaicsPI & 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 productproduct ofof disjointdisjointtranspositionstranspositions..

We define the We define the ambientambient groupgroup as the as the subgroup of the group of all permutations subgroup of the group of all permutations of the set of the set K(n) generated by the generated by the all PI all PI movesmoves and and all all ReidemeisterReidemeister movesmoves..

( )A n

The Ambient Group The Ambient Group ( )A n

7

Knot TypeKnot Type

ι→

The Mosaic InjectionThe Mosaic Injection ( ) ( 1): n nM Mι +→

We define the We define the mosaicmosaic injectioninjection ( ) ( 1): n nM Mι +→

( ),( 1)

,

if 0 ,

otherwise

ni jn

i j

K i j nK + ≤ <=

K(n+1)K(n)

Mosaic Knot TypeMosaic 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

DefDef.. Two Two nn--mosaic knots mosaic knots and and are of are of the same the same knotknot nn--typetype, written, written

K 'K

'n k

k kK i Kι+∼

Two Two nn--mosaics and are of the same mosaics and are of the same knotknot typetype if there exists a nonif there exists a non--negative negative integer integer kk such thatsuch that

K 'K

11

22 33

8

44 55

66 77

88 99

9

1010 1111

1212 1313

1414 1515

10

1616 1717

'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. KuriyaKuriya, , On a LomonacoOn a Lomonaco--Kauffman Conjecture, Kauffman Conjecture, arXiv:0811.0710. arXiv:0811.0710.

Quantum Quantum MechanicsMechanics

Whirlwind TourWhirlwind Tour

State of a Quantum SystemState of a Quantum System

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

ψHψ

Def. A Hilbert Space is a vector space over together with an inner product such that

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 vλ λ=3) , ,u v v u=4) Cauchy seq in , ∀ 1 2, ,u u … H lim nn

u→∞

∈ H

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

The dynamic behavior of a quantum system is determined by Schroedinger’s equation.

0( )U tψ ψ=

Schroedinger’sEquation

i Htψ

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

H

0ψ

Initial State

H

Hamiltonian

Dynamic State

Observable Observable

An observable is a Hermitian operator on the state space , i.e., a linear transformation such that

Ω

† TΩ = Ω = Ω

H

Quantum MeasurementQuantum Measurement

InIn OutOut

jλ

jj

j

PPψ

ψψ ψ

=ψ BlackBoxBlackBox

MacroWorldMacroWorld

QuantumQuantumWorldWorld

EigenvalueEigenvalueObservableObservable

Q. Sys.Q. Sys.StateState

Q. Sys.Q. Sys.StateState

Pr job Pψ ψ=

j jjPλΩ = ∑wherewhere Spectral DecompositionSpectral Decomposition

PhysicalPhysicalRealityReality

PhilosopherPhilosopherTurfTurf

Ω

Quantum KnotsQuantum Knots&&

Quantum Knot SystemsQuantum Knot Systems

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

The Hilbert SpaceThe Hilbert Space of Quantum Knotsof Quantum Knots( )nK

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

K(n)The The HilbertHilbert spacespace ofof quantumquantum knotsknotsis the Hilbert space with the set of nn--mosaic knotsmosaic knots as its orthonormal basis, i.e., with orthonormal basis

( )nK

K(n)

( ): nK K K∈K(n)

+K =

2

An Example of a Quantum KnotAn Example of a Quantum Knot

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 UnitaryUnitary GroupGroup( )A n

We We identifyidentify 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

Hence, under this identification, the Hence, under this identification, the ambientambientgroupgroup becomes a becomes a discrete groupdiscrete group of of unitary transfsunitary transfs on the Hilbert space .on the Hilbert space .( )nK

( )A n

+K =

2

An Example of the Group ActionAn Example of the Group Action

+2

2R K =

2R

+K =

2

( )A n

The Quantum Knot System The Quantum Knot System ( )( ) , ( )n A nK

( ) ( ) ( )(1) ( ) ( 1), (1) , ( ) , ( 1)n nA A n A nι ι ι ι

+→ → → + →K K K

PhysicallyPhysicallyImplementableImplementable

DefDef.. A A quantumquantum knotknot systemsystem 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 of accessible unitary transformations. of accessible unitary transformations.

( )nK( )A n

( )( ) , ( )n A nK



The states of quantum system areThe states of quantum system arequantumquantum knotsknots. The elements of the ambient . The elements of the ambient group aregroup are quantumquantum movesmoves..( )A n

( )( ) , ( )n A nK

The Quantum Knot System The Quantum Knot System ( )( ) , ( )n A nK

( ) ( ) ( )(1) ( ) ( 1), (1) , ( ) , ( 1)n nA A n A nι ι ι ι

+→ → → + →K K K




Choosing an integer Choosing an integer nn 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 one turns to spaciallyspacially manipulate manipulate the quantum knot.the quantum knot.

( )A n

Quantum Knot Type Quantum Knot Type

DefDef.. Two quantum knots and are Two quantum knots and are of the of the samesame knotknot nn--typetype, written, written

1K 2K

1 2 ,nK K∼provided there is an element s.t. provided there is an element s.t. ( )g A n∈

1 2g K K=

They are of the They are of the samesame knotknot typetype, written, written

1 2 ,K K∼

1 2m m

n mK Kι ι+∼provided there is an integer provided there is an integer such that such that 0m ≥

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

2R

+K =

2

+K =

2

+2

2R K =

13

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

1K =

+2

2K =

HamiltoniansHamiltoniansof theof the

GeneratorsGeneratorsof theof 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 Kα β α β α β=

( )( ) ( )11 2 2 3 1, , ,g K K K K K Kη η−

−=

Choose a permutation so thatChoose a permutation so thatη

Hence, Hence, 1

11

1

2n

g

I

σσ

η ησ

−

−

=

00

1

0 11 0

σ =

, where, where

0

1 00 1

σ =

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

For simplicity, we choose the branch .For simplicity, we choose the branch .0s =

( )( ) ( )

0 1 1

2 2

00 02 n n

I σ σπ η η−− × −

⊗ −=

( )1 1lngH i gη η η η− −= −


( ) ( ) ( )1 0 1ln 2 12

,i s sπσ σ σ= + − ∈

Log of a matrix

The Log of a Unitary MatrixThe Log of a Unitary Matrix

Let U be an arbitrary finite rxr unitary matrix.

Moreover, there exists a unitary matrix Wwhich diagonalizes U, i.e., there exists a unitary matrix W such that

( )1 21 , , , ri i iWUW e e eθ θ θ− = ∆ …

where are the eigenvalues of U.

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

1 2, , , ri i ie e eθ θ θ…

Since , where is an arbitrary integer, we have


( ) ( )1 21ln ln( ), ln( ), ,ln( )ri i iU W e e e Wθ θ θ−= ∆ …Then

( )ln 2jij je i inθ θ π= +

jn ∈

( ) ( )11 1 2 2ln 2 , 2 , , 2r rU iW n n n Wθ π θ π θ π−= ∆ + + +…

where 1 2, , , rn n n ∈…

14


( )0

/ !A m

me A m

∞

==∑

( ) ( )

( )

( )( )( )

11

1

1

1 1

1

ln , ,lnln

ln , ,ln1

ln ln1

2 21

1

, ,

, ,

, ,

r

r

r

r r

r

W i i WU

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 π θ π

θ θ

− ∆

∆−

−

+ +−

−

=

=

= ∆

= ∆

= ∆ =

…

…

…

…

…

Since , we have

Back


Using the Using the HamiltonianHamiltonian for the for the Reidemeister Reidemeister 2 move2 move ( )2,1

↔g =

cos2tπ

sin2ti π −

2i t

eπ

and the and the initial stateinitial state

we have that the we have that the solution to Schroedinger’s solution to Schroedinger’s equationequation for time isfor time ist

ObservablesObservableswhich arewhich are

Quantum KnotQuantum KnotInvariants Invariants

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

Observable Q. Knot Invariants Observable Q. Knot Invariants

QuestionQuestion.. 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 operator on the Hilbert space . operator on the Hilbert space .

( )( ) , ( )n A nKΩ

( )nK


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

Ω

DefDef.. An observable is an An observable is an invariantinvariant ofofquantumquantum knotsknots provided for provided for all all

1U U −Ω = Ω( )U A n∈

Ω( )n W=⊕K

be a decomposition of the representation be a decomposition of the representation ( ) ( )( ) n nA n × →K K


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

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

( )( ) , ( )n A nK

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 observable.observable.

PW

into irreducible representations .into irreducible representations .

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TheoremTheorem.. Let be a quantum Let be a quantum knot system, and let be an observable knot system, and let be an observable on . on .

( )( ) , ( )n A nK

Ω( )nK


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

Then the observableThen the observable

( )

1

( ) /U A n StU U −

∈ ΩΩ∑

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


• Orbit Projectors: For each mosaicknot K, the observable

( )' ( )

' 'A n KK 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.


+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 QuantumKnot Invariant

Quantum Invariantas is usually defined

An Alternate ApproachAn Alternate Approach

Next ObjectiveNext ObjectiveWe would like to find an definition of 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.

16

ReidemeisterReidemeister Moves:Moves:

R1R1

R2R2

R3R3

R0R0 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 TaziTazi knew the answer.knew the answer.

Tazi = Tasmanian Tiger

How How doesdoes 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 IdeaKey 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 ?

WiggleA curvature

Move

WagA torsion

Move

TugA metric

move

17

Can we find an alternate set of knot moves that is much more

“physics friendly” ???

Reidemeister MovesMechanical Engineers already haveMechanical 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 EngineersKnot Theory According to Mechanical Engineers

A tug

3-Bar Prismatic Linkage

Wiggles and wags can be replaced by sequences of tugs.

Hence, for extensible knot theory, there is only one move, i.e.,

which is the same as Reidemeister’s original triangle move.

Knot Theory According to Mechanical EngineersKnot 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) knottheory, and

• Inextensible Knot Theory.

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

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 only by wiggling and wagging. (Patent pending)

Quantum KnotsQuantum Knots

LatticesLattices&&

The Cubic HoneycombThe Cubic HoneycombA Scaffolding for 3A Scaffolding for 3--SpaceSpace

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

L

31 21 2 3, , : , ,

2 2 2mm 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 3A Scaffolding for 3--SpaceSpace

19

The Cubic HoneycombThe Cubic Honeycomb

Vertices

a∈ L

Edges

E

Faces

F

We think of this honeycomb as a cell complexfor consisting of:C 3

Cubes

B

All cells of positive dimension are open cells.Open OpenOpen

Lattice KnotsLattice 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 knotsof order .

( )K

Lattice KnotsLattice Knots

Lattice Trefoil Lattice Hopf Link

Necessary Necessary InfrastructureInfrastructure

Orientation of 3Orientation of 3--SpaceSpace

We define an orientation of by selecting a right handed frame

3

3e

2e1e

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 preferredframe.

3e

2e1e

3e

2e1e

Orientation of 3Orientation of 3--SpaceSpace

The preferred frame at each lattice point.

3e

2e

1e

O

20

Color Coding ConventionsColor Coding Conventionsfor Vertices & Edgesfor 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 preferredvertex 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

( ) ( )pF a = Preferred face perpendicular to pe

( ) ( )pE a = Preferred edge parallel to pe

• 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

2e

3e

1e

( )3 ( )E a

( )2 ( )E a

( )3 ( )F a

( )1 ( )E a

( )2 ( )F a

( )1 ( )F a

Preferred Vertices, Edges, & FacesPreferred Vertices, Edges, & Faces

( ) ( )B a

a

2e

3e

1e

Drawing ConventionsDrawing 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


2 ( )e a

1( )e aa

( )( )2F a

2 ( )e a

3 ( )e aa

( )( )1F a

2 ( )e aa

( )( )3F a

1( )e a

3e

1

2e

1e

2

3

a

21

The Left and Right PermutationsThe Left and Right Permutations

Define the left and right permutationsand as

: 1,2,3 1,2,31 22 33 1

→ : 1,2,3 1,2,31 32 13 2

→

p p pe e e= × p pp

e e e= × p ppe e e= ×Ergo, ( )pe a

( )pe a

( )pe a points out of the page toward the reader

PreferredVertex

Invisiblepreferred

frame

a

( )( )pE a

( )( )pE a

( )( )pF a

When drawn in isolation, is always drawn with preferred vertex in the upper left hand corner, and with pointing out of the page.

( ) ( )pF a

( )pe a


a

( )( )pF a

( )( )pF a

( )( )pE a( )( )pE a

( )( )pE a

( )( )pF a

a

pepe

pe

Vertex TranslationVertex Translation

Let be a vertex in the lattice a L

: 2ppa a e−= +

: 2ppa a e−= −

3: 3 2ppa a e−= + ⋅

52:1 2 31 2 32 2 5 2 2a a e e e− − −= + ⋅ − ⋅ +

So for example,

Lattice Knot Lattice Knot MovesMoves

Lattice knot movesLattice knot moves

DefinitionDefinition. A lattice knot move µ (oforder ) 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 aidµ − −=

( )K ∈Kfor all

22

Lattice Knot MovesLattice Knot Moves

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

TugsTugs

( )( )pF a

( )( )1 , ,L a p

This is a local move on face ( )( )pF a

( )( ) ,a p=

( ) ( )( ) ( )1 , , ,L a p a p=

TugsTugs

( )( )pF a

( )( ) , ( )a p K =

( )

( )

if

if

otherwise

K K

K K

K

− =

− =

∪ ∩

∪ ∩

means

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

( )( )pF a

( )( ) ,a p

( )( )pF a

( )( ) ,a p

( )( )pF a

( )( ) ,a p

( )( )pF a

( )( ) ,a p

12 tugs for each cube

Tugs are Tugs are extensibleextensible local moveslocal moves

For each cube, 2 Wiggles for each preferred faceFor each cube, 2 Wiggles for each preferred face

( )( )pF a

6 wiggles per cube

( )( ) ,a p

( )( )pF a

( )( ) ,a p( )2 , ,L a p = ( )2 , ,L a p =

Wiggles are Wiggles are inextensibleinextensible local moveslocal moves

( )1 ( )F a

( ) ( )( ) ( )3 ,1, ,1L a a=

WagsWags

3e

2e1eHinge

a

HingeJoint

a

HingeJoint

( )( )1F a

2e

3ea

BottomFace

( )( )1F a

3e

2e

a

BackFace

Hinge

23

The Left and Right PermutationsThe Left and Right Permutations

Please recall that the left and rightpermutations and are defined as

: 1,2,3 1,2,31 22 33 1

→ : 1,2,3 1,2,31 32 13 2

→

p p pe e e= × p pp

e e e= × p ppe e e= ×

( )( )pF a

( )( )pF a

( )( )pE a( )( )pE a

( )( )pE a

( )( )pF a

a

pepe

pe

Preferred edges & Faces

pepe

pe

PreferredVertex a

Invisiblepreferred

frame

= Edge 0⊥

= Edge 1⊥ = Edge 2⊥

= Edge 3⊥

( )( )pF a

( )1 ( )F a

( )( ) ,1a

( )1 ( )F a

( )( ) ,1a

( )1 ( )F a

( )( ) ,1a

( )1 ( )F a

( )( ) ,1a

4 Wags for face 14 Wags for face 1

WagsWags

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 inextensibleinextensible local moveslocal moves

An Example of a Wiggle MoveAn Example of a Wiggle Move

K

( )( )( ) :1 ,3a K

1e

2e

( )( ) :1 ,3a

Wiggle

Face ( ) ( ):13F a

Vertex :1a

These are Conditional

Moves

a

24

The The Ambient Ambient GroupsGroups

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 of all lattice knots of order .

0≥

( )K

In fact, each of the above local moves, as a permutation, is the product ofdisjoint 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 of involutions that generate the of involutions that generate the above groups.above groups.

Definition. The inextensible ambient groupis 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

otherwise

K K

K K

K

− =

− =

∪ ∩

∪ ∩

Conditions

25


Observation: Each is a symbolic representation of an authentic conditionalmove

3

Moreover, each involved (OP) auto-homeomorphism is local, i.e., there exists a 3-ball D such that

3 3:h →

33 3:Dh id D D− = − → −

, i.e., a conditional orientation preserving (OP) auto-homeomorphism of . Let be a family of knots in .


Let the group of local OP auto-homeomorphisms of .

( )3OPLAH

3F

For example:( )K The family of lattice knots

S The family of finitely piecewise smooth (FPWS) knots in . 3

3


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 →

F

such that( )K K KΦ ∈ ∀ ∈F F

Let be the space ofall LAC moves for the family .

( )3OPLAH

F

F


Let be the space of all LACmoves for the family .

( )3OPLAH

F

F

Define a multiplication ‘ ‘ as follows:i

( ) ( ) ( )( )

3 3 3

', 'OP OP OPLAH LAH LAH× →

Φ Φ Φ Φi

F F F

as ( ) ( )' 'K K KK ΦΦ Φ = Φ Φi

where ‘ ‘ denotes the composition of functions.


Proposition. is a monoid. ( )( )3 ,OPLAH iF

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

( )3: OPLAHΓ 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 ,OPLAH iF

3

RefinementRefinement

26

The refinement injectionThe refinement injection

( ) ( 1): +→K KQDef. We define the refinement injection

from lattice knots of order to lattice knots of order as1+

( )

( ) ( 1)

3( ) ( ) :

1 ( )

:

( ), ( )p

pp p

a p E a

K E a E a

+

∈ =

→K KQ

∪ ∪ ∪L

( ) ( ):13F a1e

2e

2

[ ]( )

*,

F x

a f

δ

δ

[ ]( )

*,

F x

a f

δ

δ

The refinement injectionThe refinement injection

An example:

Q

( )K ( 1)+K

Conjectured Refinement Conjectured Refinement MonomorphismMonomorphism

We conjecture the existence of a refinement monomorphism

1: +Λ → ΛQ

which preserves the action

( )( ) ( )

,g K gKΛ × →K K

i.e., with the property( ) ( ) ( )g K gK=Q Q Q

In fact, we have a construction which we believe is such a monomorphism.

( )( )?

( ) ,1a =Q

The Refinement Morphism ??? The Refinement Morphism ???

1e

2e

( ) ( )1F a

( ) ( ) ( ) ( )( 1) :23 ( 1) :2 ( 1) :3 ( 1),1 ,1 ,1 ,1a a a a+ + + +

1: +Λ → ΛQ

^ ^^

1^a b b ab−=

Knot TypeKnot Type

27

Lattice Knot TypeLattice Knot Type

Two lattice knots and in are of the same -type, written

1K 2K( )K

1 2~K K

provided there is an element such that g∈Λ1 2gK K=

They are of the same knot type, written1 2~K K

provided there is a non-negative integer such that

m

1 2~m m

mK K

+Q Q

Inextensible Lattice Knot TypeInextensible Lattice Knot Type

Two lattice knots and in are of the same inextensible -type, written

1K 2K( )K

1 2K K≈

provided there is an element such that g∈Λ1 2gK K=

They are of the same inextensible knot type, written

1 2K K≈

provided there is a non-negative integer such that

m

1 2m m

mK K

+≈Q Q

nn--Bounded Lattices, Bounded Lattices, Lattice knots, andLattice knots, and

Ambient GroupsAmbient Groups

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

nn--Bounded LatticesBounded Lattices

, :n a a n∞

= ∈ ≤L L

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

n

where ( )max j ja a∞ =

We also have

,nC The corresponding cell complex

,jnC The corresponding j-skeleton

nn--Bounded Lattice Knots & Bounded Lattice Knots & Ambient GroupsAmbient Groups

( , ) ( ),

nn= ∩K K L

set of n-bounded latticeknots of order

,,

nnΛ = Λ

C

,

,n

nΛ = ΛC

Ambient group of order ( ),n

Inextensible Ambient groupof order ( ),n

28

nn--Bounded Lattice Knots & Bounded Lattice Knots & Ambient GroupsAmbient Groups

We also have the injection( , ) ( , 1): n nι +→K K

( , ) ( , 1): n nι +Λ → Λ( , ) ( , 1)

:n n

ι+

Λ → Λ

and the monomorphisms

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

nΛ → Λ → → Λ →K K K

We thus have a nested sequence of lattice knot systems

and

nn--Bounded Lattice Knot TypeBounded Lattice Knot TypeTwo lattice knots and in are said to be of the same lattice knot

-type, written

1K 2K( , )nK

,ng ∈ Λ

( ),n1 2~n

K K

provided there is an element such that

1 2gK 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 nK Kι ι+

+Q Q

nn--Bounded Lattice Knot TypeBounded Lattice Knot Type

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

1 2

nK K≈ 1 2K K≈

,nΛ

andQuantum KnotsQuantum Knots

&&Quantum Knot SystemsQuantum Knot Systems

The Hilbert SpaceThe Hilbert Space of Quantum Knotsof Quantum Knots( , )nK

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

Recall that is the set of all latticeknots of order .

( , )nK

( , )n

The The HilbertHilbert spacespace ofof quantumquantum knotsknotsis the Hilbert space with the set of lattice knots of order as its orthonormal basis, i.e., with orthonormal

( , )nK( , )nK

( , )n

( , ): nK K ∈K


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

,nL

Two available cans of paint

“Solid” Red

“Hollow” Gray

An Edge

A NonA Non--EdgeEdge

Set of all 2-colorings of edges of ,nL

Set of all lattice graphs in

( , )nK

,nLIdentification

29


=E 2-D Hilbert space with orthonormal basis1 =0 =

Non-Edge Existent Edge“Hollow” Gray “Solid” Red

Edge Coloring Space E

Hilbert Space of Lattice Graphs in ( , )nG ,nL

( )( ),

( , )

n

n

E Edges∈

= ⊗L

G E

Quantum KnotsQuantum KnotsHilbert Space of Lattice Graphs in ( , )nG ,nL

( )( ),

( , )

n

n

E Edges∈

= ⊗L

G E

( )( ) ( )

( ),

( ) ( ),, | : ,

n

nE Edges

E c E c Edges∈

→ ⊗

L

L Gray Red

,lattice graph i| n nG G L

Orthonormal basis is:

which is identified with


( )( ),

( , )

n

n

E Edges∈

= ⊗L

G E

,lattice graph i| n nG G L

Orthonormal basis is:


( )( ),

( , )

n

n

E Edges∈

= ⊗L

G E

,lattice graph i| n nG G LOrthonormal basis is:

Hilbert Space of quantum knots( , )nK

( , )n =K Sub-Hilbert space of with orthonormal basis

( , )nG

( , )| kK K ∈K

+K =

2

An Example of a Quantum KnotAn 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.

,ng∈Λ

The Ambient Group as a The Ambient Group as a UnitaryUnitary GroupGroup,nΛ

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

,ng∈Λ

( , ) ( , )n n

K gK→K K

Hence, under this identification, the Hence, under this identification, the ambientambientgroupgroup becomes a becomes a discrete groupdiscrete group of of unitary transfsunitary transfs on the Hilbert space .on the Hilbert space .( , )nK

,nΛ

30

An Example of the Group ActionAn Example of the Group Action

K =

,nΛ

+2

( )( )( ) :1 ,3a K =

1e

2e

( )( ) :1 ,3a

Wiggle

+2

Face ( ) ( ):13F a

Vertex :1aThe Quantum Knot System The Quantum Knot System ( )( , )

,,nnΛK

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

n n

ι ι ι ι+

+Λ → → Λ → Λ →K K K


DefDef.. A A quantumquantum knotknot systemsystem 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 of accessible unitary transformations. of accessible unitary transformations.

( , )nK,nΛ

( )( , ),,nnΛK



The states of quantum system areThe states of quantum system arequantumquantum knotsknots. The elements of the ambient . The elements of the ambient group aregroup are quantumquantum movesmoves..( )A n

( )( , ),,nnΛK

The Quantum Knot System The Quantum Knot System ( )( , ),,nnΛK

( ) ( ) ( )( ,1) ( , ) ( , 1), , , 1, , ,n nn n n

ι ι ι ι+

+Λ → → Λ → Λ →K K K




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.

,nΛ

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

Quantum Knot Type Quantum Knot Type DefDef.. Two quantum knots and are Two quantum knots and are of the of the samesame knotknot --typetype, written, written

1K 2K

1 2 ,n

K K∼provided there is an element provided there is an element s.ts.t. . ,ng∈Λ

1 2g K K=

They are of the They are of the samesame knotknot typetype, written, written

1 2 ,K K∼

'' ' ' '

1 2'

n nn nK Kι ι

+

+∼Q Q

provided there are integer provided there are integer such that such that ', ' 0n ≥

( ),n

K =

+2

( )( )( ) :1 ,3a K =

1e

2e

( )( ) :1 ,3a

Wiggle

+2

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

1K =

+2

2K =

31

HamiltoniansHamiltoniansof theof the

GeneratorsGeneratorsof theof the

Ambient Group Ambient Group


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

,ng∈Λ

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

r rg K K K K K Kα β α β α β=

( )( ) ( )11 2 3 4 1, , ,r rg K K K K K Kη η−

−=

Choose a permutation so thatChoose a permutation so thatη

Hence, Hence,

( )

1

11

1

, 2d n r

g

I

σσ

η ησ

−

−

=

00

1

0 11 0

σ =

, where, where

( ) ( )( , ), dim nd n = K

0

1 00 1

σ =

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

For simplicity, we always choose the branch .For simplicity, we always choose the branch .0s =

( )( )( ) ( )( )

0 1 1

, 2 , 2

00 02

r

d n r d n r

I σ σπ η η−− × −

⊗ −=

( )1 1lngH i gη η η η− −= −


( ) ( ) ( )1 0 1ln 2 12

,i s sπσ σ σ= + − ∈

Matrix log def


Let U be an arbitrary finite rxr unitary matrix.

Moreover, there exists a unitary matrix Wwhich diagonalizes U, i.e., there exists a unitary matrix W such that

( )1 21 , , , ri i iWUW e e eθ θ θ− = ∆ …

where are the eigenvalues of U.

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

1 2, , , ri i ie e eθ θ θ…

Since , where is an arbitrary integer, we have


( ) ( )1 21ln ln( ), ln( ), ,ln( )ri i iU W e e e Wθ θ θ−= ∆ …Then

( )ln 2jij je i inθ θ π= +

jn ∈

( ) ( )11 1 2 2ln 2 , 2 , , 2r rU iW n n n Wθ π θ π θ π−= ∆ + + +…

where 1 2, , , rn n n ∈…


( )0

/ !A m

me A m

∞

==∑

( ) ( )

( )

( )( )( )

11

1

1

1 1

1

ln , ,lnln

ln , ,ln1

ln ln1

2 21

1

, ,

, ,

, ,

r

r

r

r r

r

W i i WU

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 π θ π

θ θ

− ∆

∆−

−

+ +−

−

=

=

= ∆

= ∆

= ∆ =

…

…

…

…

…

Since , we have

Back

32

Hamiltonians for Hamiltonians for ,nΛ

Using the Using the HamiltonianHamiltonian for the for the wiggle movewiggle move

cos2tπ

sin2t

iπ −

2i t

eπ

and the and the initial stateinitial state

we have that the we have that the solution to Schroedinger’s solution to Schroedinger’s equationequation for time isfor time ist

( )( ) :1 ,3a =

1e

2e ( ) ( ):13F a

ObservablesObservableswhich arewhich are

Quantum KnotQuantum KnotInvariants Invariants


QuestionQuestion.. 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 operator on the Hilbert space . operator on the Hilbert space .

( )( , ),,nnΛK

Ω( , )nK


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

Ω

DefDef.. An observable is an An observable is an invariantinvariant ofofquantumquantum knotsknots provided for provided for all all

1U U −Ω = Ω,nU ∈Λ

Ω

( , )nr

rW=⊕K

be a decomposition of the representation ( , ) ( , )

,n n

nΛ × →K K


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

( )( , ),,nnΛKTheorem.. Let be a quantum

knot system, and let

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

rPrrW

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 on .

( )( , ),,nnΛK

Ω( , )nK


( ) 1( ) :St U A n U U −Ω = ∈ Ω = Ω

Then the observable

( ),

1

/nU StU U −

∈Λ Ω

Ω∑is a quantum knot invariant, where the above sum is over a complete set of coset representatives of in . ( )St Ω ,nΛ

33


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

( ) ( ) ( ) ( )3 3

( ) ( ) ( ) : ( ) :

1 1

p pp p p p

p pF a F a F a F a

= =

Ω = ∂ ∂ + ∂ ∂∑ ∑

where denotes the boundary of the face .

( )( )pF a∂

( )( )pF a

( , )nK Future DirectionsFuture Directions

&&

Open QuestionsOpen Questions

Future Directions & Open QuestionsFuture Directions & Open Questions

•• What is the structure of the ambient groupsWhat 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 CoxeterCoxeter groups?groups?

Λ ,nΛ ,nΛ

•• 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 ?


• Unlike classical knots, quantum knots canexhibit the non-classical behavior of quantum superposition and quantum entanglement. Are quantum and topologicalentanglement related to one another ? If so, how ?


•• For each lattice knot For each lattice knot KK, let denote, let denoteits Jones polynomial.its Jones polynomial. We define the We define the JonesJonesobservableobservable asas

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

( )KV t

( ) ( )( )n

KK

V t V t K K∈

= ∑K(n)

Can it be used to create a different quantum algorithm for the Jones polynomial ?


•• What is gained by extending the definition of quantum knot observables to POVMs ?

•• What is gained by extending the definition of quantum knot observables to mixed ensembles ?

34


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 observable for the lattice number?

( )0,n=


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 .0ε >

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 QuestionsFuture Directions & Open Questions•• Can quantum knot systems be used to model

and predict the behavior ofQuantum vortices in supercooled helium 2 ?

Fractional charge quantification that ismanifest in the fractional quantum Halleffect

Quantum vortices in the Bose-EinsteinCondensate

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 xx∂ ∂⊗∂ ∂

( ) 1 2

*

dx dxx

( ) 1 2

1 2, ,0 dx dxx x1 0..1x = 2 0..1x =

[ ]( )

*,

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∪ ∪ ∪

( )PVa

Half Closed Cube

PreferredVertex

The Preferred Vertex (PV) MapThe Preferred Vertex (PV) Map −

Since the half closed cubes form a partition of 3-space , we have the preferred vertex map3

( ) 3:PV → 31 21 2 3, , : , ,

2 2 2mm m m m m = ∈

L

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 on

FPS =F

: FPSF →FFPSF

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

*, , , , ,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 xF x

a f

δ

δ−

→∞

− =

[ ]( )

( ) ( ) [ ]( )

( )

*

2

*

,

2,lim

F a f PV x F xF x

a f

δ

δ−

→∞

− =


[ ]( )

( ) ( ) [ ]( )

( )

*

2

*

,

2,lim

F a f PV x F xF x

a f

δ

δ−

→∞

− =

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

: FPSF →F


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