Quantum Knots ??? Quantum Knots - Inspiring Innovationlomonaco/powerpoint... · LomonacoLomonaco, Samuel J., Jr.,, Samuel J., Jr., The modern legacies of Thomson's atomic vortex theory

1

Quantum Quantum Knots ???Knots ???

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

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

Quantum KnotsQuantum Knots

LomonacoLomonaco LibraryLibrary

Two papers on Two papers on Quantum Knots Quantum Knots can be found in can be found in this book.this book.

Throughout this talk:

“Knot” means either a knot or a link

Lomonaco and Kauffman,Lomonaco and Kauffman, Quantum Knots and Quantum Knots and Lattices,Lattices, AMS PSAPM/68, (2010), 209AMS PSAPM/68, (2010), 209--276276

This talk is based on the papers:This talk is based on the papers:

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.

2

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

LomonacoLomonaco, Samuel J., Jr.,, 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.

All the papers on Quantum Knots can be found All the papers on Quantum Knots can be found on Quanton Quant--Ph and on the website.Ph and on the website.

PowerPoint slides can be found at: PowerPoint slides can be found at: www.csee.umbc.edu/~lomonaco/Lectures.htmlwww.csee.umbc.edu/~lomonaco/Lectures.html

www.csee.umbc.edu/~lomonaco

Classical Vortices in PlasmasClassical Vortices in Plasmas

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.

PreamblePreamble

Knots Naturally Arise in the Knots Naturally Arise in the QuantumQuantum World as Dynamical ProcessesWorld as Dynamical Processes

Examples of dynamical knots in quantum physics:Examples of dynamical knots in quantum physics:KnottedKnotted vorticesvortices

•• 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

Reason for current intense interest:Reason for current intense interest:Topology Is a Natural Obstruction to Topology Is a Natural Obstruction to DecoherenceDecoherence

Thinking Outside the BoxThinking Outside the Box

Knot TheoryKnot Theory

Quantum MechanicsQuantum Mechanics

is a tool for exploringis a tool for exploring

3

Quantum Topology Quantum Topology

•The ultimate objective is to create and to investigate mathematical objects that can be physically implemented in a quantum physics lab.

Quantum Physics

•The objective of this work is to do topology in such a way that it is intimately related to quantum physics

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.

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

At least 3 equiv ways to create Quantum Knots1) Mosaic Construction:• Advantages:

• Disadvantages:

2) Lattice Construction:• Advantages:- Geometry of 3-space transparent- Moves become variational derivatives- Fits seamlessly into physics

- Easiest least technical approach

- Geometry of 3-space not transparent

• Disadvantages:- Fraught with technicalities

Lattice Approach

• With fear and trepidation, we must abandonthe Reidemeister moves, and replace them with the truly 3-D moves:

Wiggle, Wag, & Tug

so named because these moves mimic how a dog moves its tail.

Curvaturemove

Torsionmovemove

Metricmove

4

At least 3 equiv ways to create Quantum Knots

3) Smooth curves in construction:

• Advantages:

• Disadvantages:

- No longer working with PL knots

- Many technical obstacles

3

- All the advantages of lattice approach

KnotTheory

QuantumMechanics

GroupRepresentation

Theory=

FormalRewritingSystem

=

FormalRewritingSystem

= GroupRepresentation

OutlineOutline

Mosaic KnotsMosaic Knots

Part 1Part 1

??????

Mosaic KnotsMosaic Knots

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

Definition of an nDefinition of an n--MosaicMosaic

An An nn--mosaicmosaic is an matrix of tiles, with is an matrix of tiles, with rows and columns indexed rows and columns indexed

n n0,1, , 1n

An example of a 4An example of a 4--mosaicmosaic

5

Tile Connection PointsTile Connection Points

A A connectionconnection pointpoint of a tile is a midpoint of a tile is a midpoint of an edge which is also the endpoint of a of an edge which is also the endpoint of a curve drawn on a tile. For example,curve drawn on a tile. For example,

00ConnectionConnection

PointsPoints


PointsPoints


PointsPoints

Contiguous TilesContiguous Tiles

Two tiles in a mosaic are said to be Two tiles in a mosaic are said to be contiguouscontiguousif they lie immediately next to each other in if they lie immediately next to each other in either the either the same rowsame row or the or the same columnsame column..

ContiguousContiguous Not ContiguousNot Contiguous

Suitably Connected TilesSuitably Connected Tiles

A tile in a mosaic is said to be A tile in a mosaic is said to be SuitablySuitablyConnectedConnected if all its connection points touch the if all its connection points touch the connection points of contiguous tiles. For connection points of contiguous tiles. For example,example,

SuitablySuitablyConnectedConnected

Not SuitablyNot SuitablyConnectedConnected

KnotKnot MosaicsMosaics

A A knotknot mosaicmosaic is a mosaic with all tiles suitably is a mosaic with all tiles suitably connected. For example,connected. For example,

Knot Knot 44--MosaicMosaicNonNon--Knot Knot 44--MosaicMosaic

Figure Eight KnotFigure Eight Knot 55--MosaicMosaic Hopf LinkHopf Link 44--MosaicMosaic

6

Borromean RingsBorromean Rings 66--MosaicMosaic NotationNotation

( )nM Set of Set of nn--mosaicsmosaics

( )nK Subset of knot Subset of knot nn--mosaicsmosaics

Planar Isotopy Planar Isotopy MovesMoves

NonNon--Determistic TilesDetermistic Tiles

We use the following tile symbols to denote We use the following tile symbols to denote one of two possible tiles:one of two possible tiles:

For example, the tile denotes eitherFor example, the tile denotes either

oror

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

7

Terminology: kTerminology: k--Submosaic MovesSubmosaic Moves

All of the PI moves are examples of All of the PI moves are examples of 22--submosaic moves. I.e., each PI submosaic moves. I.e., each PI move replaces a move replaces a 22--submosaic by another submosaic by another 22--submosaicsubmosaic

DefDef.. A A kk--submosaicsubmosaic movemove on a mosaic on a mosaic MM is a mosaic move that replaces one is a mosaic move that replaces one kk--submosaic in submosaic in MM by another by another kk--submosaic. submosaic.

For example, For example, 1P


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--k+1)k+1)22 possible moves on an possible moves on an nn--mosaicmosaic


Each PI move acts as a local transformation Each PI move acts as a local transformation on an on an nn--mosaic whenever its conditions are mosaic whenever its conditions are met. If its conditions are not met, it acts met. If its conditions are not met, it acts as the identity transformation. as the identity transformation.

Ergo, each Ergo, each PI movePI move is a is a permutationpermutation of the of the set of all knot set of all knot nn--mosaicsmosaics ( )nK

In fact, each In fact, each PI movePI move, as a permutation, is , as a permutation, is a a productproduct ofof disjointdisjoint transpositionstranspositions..

Reidemeister Reidemeister MovesMoves

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

2R

1R '1R

'2R

''2R

'''2R

Reidemeister 1 MovesReidemeister 1 Moves

Reidemeister 2 MovesReidemeister 2 Moves

More NonMore Non--Deterministic TilesDeterministic Tiles

We also use the following tile symbols to We also use the following tile symbols to denote one of two possible tiles:denote one of two possible tiles:

oror

For example, the tile denotes eitherFor example, the tile denotes either

8

Synchronized NonSynchronized Non--Determistic TilesDetermistic Tiles

Nondeterministic tiles labeled by the same Nondeterministic tiles labeled by the same letter are letter are synchronizedsynchronized::

Reidemeister 3 (R3) Moves on MosaicsReidemeister 3 (R3) Moves on Mosaics

3R '3R

'''3R

''3R

( )3

ivR( )3vR

Just like each PI move, each Just like each PI move, each R moveR moveis a permutation of the set of all is a permutation of the set of all knot knot nn--mosaicsmosaics ( )nK

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

In fact, each In fact, each R moveR move, as a permutation, is , as a permutation, is a a productproduct ofof disjointdisjoint transpositionstranspositions..

We define the We define the ambientambient isotopyisotopy groupgroupas the subgroup of the group of as the subgroup of the group of

all permutations of the set all permutations of the set generated by the generated by the all PI movesall PI moves and and all all Reidemeister movesReidemeister moves..

( )A n( )nK

The Ambient Group The Ambient Group ( )A n

Knot TypeKnot Type

The Mosaic InjectionThe Mosaic Injection ( ) ( 1): n nM M

We define the We define the mosaicmosaic injectioninjection ( ) ( 1): n nM M

( ),( 1)

,

if 0 ,

otherwise

ni jn

i j

M i j nM

9

Mosaic Knot TypeMosaic Knot Type

'n

M Mprovided 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 M 'M

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

M 'M

'n k

k kM i M

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

M 'M

11

22 33

44 55

10

66 77

88 99

1010 1111

11

1212 1313

1414 1515

1616 1717

'n

K K

12

Oriented Oriented MosaicsMosaics

There are There are 2929 oriented tiles, and oriented tiles, and 99 tiles up tiles up to rotation. Rotationally equivalent tiles to rotation. Rotationally equivalent tiles have been grouped together.have been grouped together.

Oriented Mosaics and Oriented Knot TypeOriented Mosaics and Oriented Knot Type

In like manner, we can use the following In like manner, we can use the following oriented tiles to construct oriented tiles to construct oriented mosaicsoriented mosaics, , oriented mosaic knotsoriented mosaic knots, and , and oriented knot typeoriented knot type

Quantum KnotsQuantum Knots&&

Quantum Knot SystemsQuantum Knot Systems

Part 2Part 2 Let be the Let be the 1111 dimensional Hilbert space dimensional Hilbert space with orthonormal basis labeled by the tiles with orthonormal basis labeled by the tiles

H

We define the We define the HilbertHilbert spacespace ofof nn--mosaicsmosaicsas as

( )nM2 1

( )

0

nn

k

M H

This is the Hilbert space with induced This is the Hilbert space with induced orthonormal basisorthonormal basis

2 1

( )0: 0 ( ) 11

n

kkT k

0T 1T 2T 3T 4T 5T 6T 7T 8T 9T 10T

The Hilbert SpaceThe Hilbert Space of nof n--mosaicsmosaics( )nM

is identified with the is identified with the 33--mosaic labeled ketmosaic labeled ket

For example, in the For example, in the 33--mosaic Hilbert space mosaic Hilbert space , the basis ket , the basis ket ( 3)M

We identify each basis ket withWe identify each basis ket witha ket labeled by an a ket labeled by an nn--mosaic using mosaic using row major orderrow major order. .

2 1

( )0

n

kkT

M M

2 5 4 9 2 1 5 8 3T T T T T T T T T

2 5 4

9 2 1

5 8 3

T T T

T T T

T T T

The Hilbert SpaceThe Hilbert Space of nof n--mosaicsmosaics( )nM

H

H

H H H H

H H H H

H H H

H H H

Identification via Row Major OrderIdentification via Row Major OrderLet be the Let be the 1111 dimensional Hilbert space dimensional Hilbert space with orthonormal basis labeled by the tiles with orthonormal basis labeled by the tiles

H

0T 1T 2T 3T 4T 5T 6T 7T 8T 9T 10T

2nH

SelectSelectBasisBasis

ElementElement

( , )0 ,

k i ji j n

T

ConstructConstructMosaic SpaceMosaic Space

Row Major OrderRow Major Order

13

The The HilbertHilbert spacespace ofof quantumquantum knotsknots is is defined as the subdefined as the sub--Hilbert space of Hilbert space of spanned by all orthonormal basis elements spanned by all orthonormal basis elements labeled by knot labeled by knot nn--mosaicsmosaics. .

( )nK( )nM

The Hilbert SpaceThe Hilbert Space of Quantum Knotsof Quantum Knots( )nKQuantum KnotsQuantum Knots

We define the We define the HilbertHilbert spacespace ofof nn--mosaicsmosaicsas as

( )nM2 1( )

0

nn

k

M H

This is the Hilbert space with induced This is the Hilbert space with induced orthonormal basisorthonormal basis

2 1 2( )0

: 0n

kkT n

We identify each basis element We identify each basis element with the mosaic labeled ket via the bijection with the mosaic labeled ket via the bijection

2 1

( )0

n

kkT

M,i jT M

where where /

/

i n

j n n

andand ni j

Row major orderRow major order

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.

( )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 n

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

14

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

(1) ( ) ( 1), (1) , ( ) , ( 1)n nA A n A n

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 “knobs” one turns to spacially manipulate the “knobs” one turns to spacially 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 Kprovided 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 K 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

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 K

11 2 3 3 1, , ,g K K K K K K

Choose a permutation so thatChoose a permutation so that

Hence, Hence, 1

1

1

1

2n

g

I

00

1

0 1

1 0

, where, where

15

0

1 0

0 1

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

0

0 02 n n

I

1 1lngH i g


1 0 1ln 2 12

,i

s s


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

g

cos2

t

sin2

ti

2

i t

e

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

Some Some Miscellaneous Miscellaneous

Unitary Unitary Transformations Transformations

Not inNot in( )A n

Misc. Transformations Misc. Transformations

,i j

ij

,i j

1

, 0

n

i j

The crossing tunneling transformationThe crossing tunneling transformation

The mirror image transformationThe mirror image transformation

Misc. Transformations Misc. Transformations

,i j

ij

,i j

ij

The hyperbolic transformationThe hyperbolic transformation

The elliptic transformationThe elliptic transformation

ObservablesObservableswhich arewhich are

Quantum KnotQuantum KnotInvariants Invariants

16

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 nK

( )nK


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

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

1U U ( )U A n

( )n WK

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 quantum knot for the subspace is quantum knot observable.observable.

PW

into irreducible representations .into irreducible representations .

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

( )nK


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

St

1( ) :St U A n U U

Then the observableThen the observable

1

( ) /U A n St

U U

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 ( )A n


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

Future DirectionsFuture Directions

&&

Open QuestionsOpen Questions

17

Future Directions & Open QuestionsFuture Directions & Open Questions

•• What is the structure of the ambient groupWhat is the structure of the ambient groupand its direct limit ?and its direct limit ?( )A n lim ( )A A n

Can one find a presentation of this group ?Can one find a presentation of this group ?

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


•• How does one find a quantum observable for How does one find a quantum observable for the Jones polynomial ? the Jones polynomial ?

•• How does one create quantum knot How does one create quantum knot observables that represent other knot observables that represent other knot invariants such as, for example, the Vassiliev invariants such as, for example, the Vassiliev invariants ?invariants ?

This would be a family This would be a family of observables parameterized by points on the of observables parameterized by points on the unit circle in the complex plane. unit circle in the complex plane. Does this Does this approach lead to an algorithmic improvement approach lead to an algorithmic improvement to the quantum algorithm created by to the quantum algorithm created by AharonovAharonov, Jones, and Landau ? , Jones, and Landau ?


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

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


DefDef.. We define the We define the mosaicmosaic numbernumber of a knot of a knot kk as the smallest integer as the smallest integer nn for which for which kk is is representable as a knot representable as a knot nn--mosaic.mosaic.

•• The mosaic number of the trefoilThe mosaic number of the trefoilis is 44. .

•• Is the mosaic number related to theIs the mosaic number related to thecrossing number of a knot?crossing number of a knot?

In general, how does oneIn general, how does onecompute the mosaic number of acompute the mosaic number of aknot?knot?

Future Directions & Open QuestionsFuture 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 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.

UMBCUMBCQuantum Knots Quantum Knots Research LabResearch Lab

Skip to lattices

18

Quantum Knots Research LabQuantum Knots Research Lab Weird !!!Weird !!!

STOPSTOP 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.

ReidemeisterReidemeister Moves:Moves:

R1R1

R2R2

R3R3

R0R0

19

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

Zawitz’s Tangle

Moves only by Wagging

20

Lomonaco’s Bendangle

Moves only by Wiggling – Patent Pending

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

“physics friendly” ???

Reidemeister Moves

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

Yes, when Yes, when TaziTazi moved her tail, she naturally moved her tail, she naturally understood how a curve can move in 3understood how a curve can move in 3--space !space !

She had a keen understanding of She had a keen understanding of differential differential geometrygeometry..

Skip to mechanical

Differential Geometry: The Frenet FrameDifferential Geometry: The Frenet Frame

Each point of a curve in 3Each point of a curve in 3--space is naturally space is naturally associated with a 3associated with a 3--frame, called the frame, called the FrenetFrenet frameframe. .

T = T = tangenttangent

B = B = binormalbinormal = = TxNTxN

N=N=normalnormal


Each point of a curve in 3Each point of a curve in 3--space is naturally space is naturally associated with a 3associated with a 3--frame, called the frame, called the FrenetFrenet frameframe. .



N=N=normalnormal




N=N=normalnormal

•• A curve stretches or contracts along its tangent T

•• A curve bends by rotating about B – as measured by its curvature •• A curve twists by rotating about N – as measured by its torsion

21

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

Clues from Mechanical EngineeringClues from Mechanical EngineeringLinkage = Inextensible bars (i.e., rods) connected by joints

Joints:

• Planar • Spherical

• Slider

Clues from Mechanical EngineeringClues from Mechanical Engineering

Mechanism = a linkage with 1 degree offreedom

MechanismsMechanisms

4-Bar LinkageAll Joints Planar

Fixed

4-th Bar

Since endpoints fixed, this is a local move on linkages. The rest of the linkage is untouched

This is a local curvature move, taking place in a fixed plane. We call it a wiggle.


3-Bar Linkage

All Joints Spherical

3-rd Bar


This is a local torsion move, locally twisting the linkage into a new plane. We call it a wag.

Fixed


4-Bar SliderAll

Joints Planar except Slider

4-th Bar


This is a local expansion/contraction move, taking place in a fixed plane. We call it a tug.

Fixed

22

Translating M.E. into Knot TheoryTranslating M.E. into Knot TheoryDefinition: Two piecewise linear (PL) knots K1 and K2 are said to be of the sameextensible knot type, written K1 ~ K2, provided one can be transformed into the other by a finite sequence of the following local moves:

2) A wiggle:

3) A wag:

1) A tug:

Translating M.E. into Knot TheoryTranslating M.E. into Knot Theory

Using the methods found in Reidemeister’s proof of the completeness of the Reidemeister moves, we have:

Theorem: Wiggles and wags can be expressed as sequences of tugs.

In fact, Reidemeister’s fundamental move was essentially a tug.

So why bother with wiggles and wags ?

Why Wiggle & Wag?Why Wiggle & Wag?

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.

My reason is that, while investigating electromagnetic knots, the knot theoretic tools I needed to study knots that naturally arise in physics were simply not available.

What was needed was a knot theory for inextensible knots. Reidemeister’s moves, which are essentially derived from the tug move, are simply NOT inextensible moves.

Inextensible Knot TheoryInextensible Knot Theory

Definition: Two piecewise linear (PL) knots K1 and K2 are said to be of the sameinextensible knot type, written

K1 K2 ,provided one can be transformed into the other by a finite sequence of wiggles and wags.

Proposition. Let K1 and K2 be PL knots. Then

1 2

1 2

1 2

~

&

K K

K K

K K


Proposition. Let K1 and K2 be PL knots. Then

1 2

1 2

1 2

~

&

K K

K K

K K

So it would seem that we have gained NOTHINGby creating inextensible knot theory !!!

But think again !But think again !


By working with this modified definition of knot type,

• We have lost none of the structure of classical knot theory.

• But we have succeeded in incorporating more of the geometry of 3-space.

23


Because of this modified definition, we will be able to:

• Create infinitesimal knot moves

• Create knot move differential forms

• Take variational derivatives with respect to these infinitesimal moves

• And much more

Lattice KnotsLattice Knots

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 2

mm mm 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

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


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

24


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

2e

1e

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

2e

1e

Orientation of 3Orientation of 3--SpaceSpace

The preferred frame at each lattice point.

3e

2e

1e

O

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 its preferred vertex, we use the notation

( ) ( )B B a

Skip to moves

25

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 face 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

The Left and Right PermutationsThe Left and Right Permutations

Define the left and right permutationsand 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 pe e e p pp

e e e p ppe e e Ergo, ( )pe a

( )p

e 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

26

( )( )pF a

( )( )pF a

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

( )( )p

E a

( )( )p

F a

a

pep

e

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,

( ) ( ) ( )( ) ( ) ( )B a a E a F a

a

Half Closed Cube

PreferredVertex

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

3

1 2 3 1 2 3

:

, , 2 2 ,2 2 ,2 2x x x x x x x

L

Lattice Knot Lattice Knot MovesMoves

Skip to tug


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 latticeknots of order .

( )K


Lattice Trefoil Lattice Hopf Link

27

Lattice knot movesLattice knot moves

DefinitionDefinition. A lattice knot move (oforder ) is a bijection

( ) ( ): K K

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

( )( )B a

( ) ( )( ) ( ) ( )K B a K B a

( )K Kfor all

Lattice Knot MovesLattice Knot Moves

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

TugsTugs

( )pF a

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

This is a local move on face ( )pF a

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

TugsTugs

( )pF a

( ) ,a p

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

28

( )1 ( )F a

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

WagsWags

3e

2e

1eHinge

a

HingeJoint

a

HingeJoint

( )1F a

2e

3ea

BottomFace

( )1F a

3e

2e

a

BackFace

Hinge

Skip to wag

The Left and Right PermutationsThe Left and Right Permutations

Please recall that the left and rightpermutations and are defined as

: 1,2,3 1,2,3

1 2

2 3

3 1

: 1,2,3 1,2,3

1 3

2 1

3 2

p p pe e e p pp

e e e p ppe e e

( )( )pF a

( )( )pF a

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

( )( )p

E a

( )( )pF a

a

pep

e

pe

Preferred edges & Faces

pep

e

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

Skip to example

29

An Example of a Wiggle MoveAn Example of a Wiggle Move

K

( ) :1 ,3a K

1e

2e

( ) :1 ,3a

Wiggle

Face :1

3F a

Vertex :1a

These are Conditional

Moves

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

Definition. The inextensible ambient groupis the group generated only by wiggles

and wags of order .

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.

What Is the Ambient Group ?What Is the Ambient Group ?

Skip to knot

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

30


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

F

such that K K K 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 .

3O PLA H

F

F

Define a multiplication ‘ ‘ as follows:

3 3 3

', '

OP OP OPLAH LAH LAH

F F F

as ( )' 'K K KK

where ‘ ‘ denotes function composition.


Proposition. is a monoid. 3 ,O PLA H F

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

3: OPLAH F

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

3 ,O PLA H F

3

RefinementRefinement

31

The refinement injectionThe refinement injection

( ) ( 1): K KQ Def. We define the refinement injection

from lattice knots of order to lattice knots of order as

1

( )

( ) ( 1)

3( ) ( ) :

1 ( )

:

( ), ( )p

pp p

a p E a

K E a E a

K KQ

L

:13F a1e

2e

2

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: Q

which preserves the action

( ) ( )

,g K gK

K K

i.e., with the property g K gKQ 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: Q

Knot TypeKnot Type

Skip to qknots

32

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

1 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

mQ K Q K

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

1 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

mQ K Q K

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

Lset of n-bounded latticeknots of order

,,

nn

C

,

,n

n

C

Ambient group of order ,n

Inextensible Ambient groupof order ,n

33

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

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

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

n n

and the monomorphisms

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

n 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

,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 nn nK K

Q Q

nn--Bounded Lattice Knot TypeBounded Lattice Knot Type

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

1 2

n

K K

1 2K K

,n

andQuantum KnotsQuantum Knots

&&Quantum Knot SystemsQuantum Knot Systems


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

,nL

Identification


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

34

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


,nL

( ),

( , )

n

n

E Edges

L

G E

,lattice graph i| n nG G L

Orthonormal basis is:


,nL

( ),

( , )

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

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

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

,ng

( , ) ( , )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

,n

An Example of the Group ActionAn Example of the Group Action

K

,n

2

( ) :1 ,3a K

1e

2e

( ) :1 ,3a

Wiggle

2

Face :1

3F a

Vertex :1a

35

The Quantum Knot System The Quantum Knot System ( , ),,nnK

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

n n

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

,n

( , ),,nnK



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

( , ),,nnK

The parameters (wiggle, wag, & tug) of the ambient group are the “knobs” one turns to spacially manipulate the quantum knot.

The Quantum Knot System The Quantum Knot System ( , ),,nnK

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

K K K




Choosing integers and n is analogous to choosing respectively the thickness and the length of the rope.

,n

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

UMBC QLab

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

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 K

Q Q

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

,n

UMBC QLab

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

HamiltoniansHamiltoniansof theof the

GeneratorsGeneratorsof theof the

Ambient Group Ambient Group

36


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

,ng

1 1 2 2, , ,

r rg K K K K K K

11 2 3 4 1, , ,r rg K K K K K K

Choose a permutation so thatChoose a permutation so that

Hence, Hence,

1

1

1

1

, 2d n r

g

I

00

1

0 1

1 0

, where, where

( , ), dim nd n K

0

1 0

0 1

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

0

0 02r

d n r d n r

I

1 1lngH i g


1 0 1ln 2 12

,i

s s

Matrix log def

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 e

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 e

Since , where is an arbitrary integer, we have


1 21ln ln( ), ln( ), , ln( )ri i iU W e e e W Then

ln 2ji

j je i in

jn

11 1 2 2ln 2 , 2 , , 2r rU iW n n n W

where 1 2, , , rn n n


0

/ !A m

m

e 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

Since , we have

Back

Hamiltonians for Hamiltonians for ,n

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

cos2

t

sin2

ti

2

i t

e

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

37

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 .

( , ),,nnK

( , )nK


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

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

1U U ,nU

( , )n

rr

WK

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

,n n

n K K


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

( , ),,nnK

Theorem.. 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.,

St

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

( , ),,nnK

( , )nK


1( ) :St U A n U U

Then the observable

,

1

/nU St

U U

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


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

3 3

( ) ( ) ( ) : ( ) :

1 1

p pp p p p

p p

F a F a F a F a

where denotes the boundary of the face .

( )pF a

( )pF a

( , )nK

38

Future DirectionsFuture Directions

&&

Open QuestionsOpen Questions


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

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

,n ,n

•• 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.


• 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 ?


•• How does one find a quantum observable for the Jones polynomial ?

•• How does one create quantum knot observables that represent other knot invariants such as, for example, the Vassilievinvariants ?

This would be a family of observables parameterized by points on the circle in the complex plane. Does this approach lead to an algorithmic improvement to the quantum algorithm created by Aharonov, Jones, and Landau ?


•• 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 ?


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

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 mosaic number?

0,n

39


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 .0

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?


•• Can quantum knot systems be used to modeland predict the behavior of

Quantum vortices in supercooled helium 2 ?

Fractional charge quantification that ismanifest in the fractional quantum Hall effect

Quantum vortices in the Bose-EinsteinCondensate

Tug, Wiggle, WagTug, Wiggle, Wagareare

“Physics Friendly”“Physics Friendly”

UMBC QLab

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

1 1

*x xx

1 2

*

dx dxx

1 2

1 2, ,0dx dx

x x1 0..1x 2 0..1x

*,

F x

a f

*,

F x

a f

Variational Derivatives w.r.t. moves

*,

F x

a f

( ) ( ) ( )( ) ( ) ( )B a a E a F a

( )PV

a

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 2

mm mm m m

LUMBC QLab

40

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 F FPSF

( ) ( ) ( ) ( ) ( )

*, , , , ,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

( )

( )

*

2

*

,

2,lim

F a f PV x F xF x

a f


( )

( )

*

2

*

,

2,lim

F a f PV x F xF x

a f

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

: FPSF F


UMBCUMBCQuantum Knots Quantum Knots Research LabResearch 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 !!!

Quantum Knots Research LabQuantum Knots Research Lab

41

Weird !!!Weird !!!

Documents

Quantum Knots ??? Quantum Knots - Inspiring Innovationlomonaco/powerpoint... · LomonacoLomonaco, Samuel J., Jr.,, Samuel J., Jr., The modern legacies of Thomson's atomic vortex theory