Eureka Quantum Knots Quantum Knots Mosaics Revisited Samuel

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Eureka !!! Quantum Knots

Eureka !!! Quantum Knots

Quantum Knots & Mosaics: Revisited Samuel Lomonaco University of Maryland Baltimore County (UMBC) Email:

Quantum Knots & Mosaics: Revisited Samuel Lomonaco University of Maryland Baltimore County (UMBC) Email: Lomonaco@UMBC. edu Web. Page: www. csee. umbc. edu/~lomonaco The L-O-O-P Fund Talk given at Knot Theory Course 3 July 2020

This is joint work with Louis Kauffman

This is joint work with Louis Kauffman

This talk is based on the papers: Lomonaco and Kauffman, Quantum Knots and Mosaics,

This talk is based on the papers: Lomonaco and Kauffman, Quantum Knots and Mosaics, Journal of Quantum Information Processing, vol. 7, Nos. 2 -3, (2008), 85 -115. And related to the papers: Lomonaco and Kauffman, Quantum Knots and Lattices, AMS PSAPM/68, (2010), 209 -276 Lomonaco and Kauffman, Quantizing Braids & Other Mathematical Structures: The General Quantization Procedure, http: //arxiv. org/abs/1105. 0371

This talk was motivated by: Rasetti, Mario, and Tullio Regge, Vortices in He II,

This talk was motivated by: Rasetti, Mario, and Tullio Regge, Vortices in He II, current algebras and quantum knots, Physica 80 A, North-Holland, (1975), 2172333. Kitaev, Alexei Yu, Fault-tolerant quantum computation by anyons, http: //arxiv. org/abs/quant-ph/9707021

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

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

Intro

Intro

Quantum Mechanics is a tool for exploring Knot Theory

Quantum Mechanics is a tool for exploring Knot Theory

Objectives • To create a quantum system that simulates a closed knotted physical piece

Objectives • To create a quantum system that simulates a closed knotted physical piece of rope. • To define a quantum knot in such a way as to represent the state of the knotted rope, i. e. , the particular spatial configuration of the knot tied in the rope. • To model the ways of moving the rope around (without cutting the rope, and without letting it pass through itself. )

Rules of the Game Find a mathematical definition of a quantum knot that is

Rules of the Game Find a mathematical definition of a quantum knot that is • Physically meaningful, i. e. , physically implementable, and • Simple enough to be workable and useable.

Aspirations We would hope that this definition will be useful in modeling and predicting

Aspirations We would hope that this definition will be useful in modeling and predicting the behavior of knotted vortices that actually occur in quantum physics.

Outline Knot Theory = Formal Rewriting System = Group Representation Theory Quantum Mechanics

Outline Knot Theory = Formal Rewriting System = Group Representation Theory Quantum Mechanics

Placement Problem: Knot Theory • Ambient space • Group Orientation Preserving 1 # t

Placement Problem: Knot Theory • Ambient space • Group Orientation Preserving 1 # t n eme c a l P Placement#2 Def. Problem. When are two placements the same ? ?

Equivalent Definition Def. K 1 ~ K 2 provided there exists a continuous family

Equivalent Definition Def. K 1 ~ K 2 provided there exists a continuous family of auto-homeomorphisms i. e. , an isotopy, that continuously deforms K 1 into K 2.

Knot Projections

Knot Projections

Knot Diagram Labeled Vertex • Planar four valent graph with • Labeled vertices Labeled

Knot Diagram Labeled Vertex • Planar four valent graph with • Labeled vertices Labeled Vertex

Reidemeister Moves R 0 R 1 R 2 R 3 These are local moves

Reidemeister Moves R 0 R 1 R 2 R 3 These are local moves !

When do two Knot diagrams represent the same or different knots ? Theorem (Reidemeister).

When do two Knot diagrams represent the same or different knots ? Theorem (Reidemeister). Two knot diagrams represent the same knot type iff one can be transformed into the other by a finite sequence of Reidemester moves. This theorem has produced Gainful Employment for knot theorist for almost 100 years!

Preamble to Mosaic Knots

Preamble to Mosaic Knots

Preamble to Mosaic Knots En route to defining Q Knots, we encountered the following

Preamble to Mosaic Knots En route to defining Q Knots, we encountered the following obstacle: Obstacle: {Knot Diagrams, R Moves} = Category. But Q. M. requires that it be a group!

Preamble to Mosaic Knots Problem: {Knot Diagrams, R Moves} = Category. But Q. M.

Preamble to Mosaic Knots Problem: {Knot Diagrams, R Moves} = Category. But Q. M. requires that it be a group !!! Resolution: The Principle of Conditional Action. • An R-move acts as an identity transformation if it cannot be applied. • An R-move applied twice is the identity transformation, i. e. , Voila! , {Knot Diagrams, R Moves} = Group Hence, all R-moves are involutions

Preamble to Mosaic Knots Spin-Off 1 • Tame Knot Theory is a formal rewriting

Preamble to Mosaic Knots Spin-Off 1 • Tame Knot Theory is a formal rewriting system, i. e. , a context sensitive formal language (defined by a linear-bounded automata. ) Knots = Meaningless strings of symbols R-moves = Rewriting rules, grammatical rules

Preamble to Mosaic Knots Spin-Off 2 We have created an axiomatic system that completely

Preamble to Mosaic Knots Spin-Off 2 We have created an axiomatic system that completely captures and defines tame knot theory.

Mosaic Knots

Mosaic Knots

? ? ? Mosaic Knots

? ? ? Mosaic Knots

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

Mosaic Tiles Let denote the following set of 11 symbols, called mosaic (unoriented) tiles: Please note that, up to rotation, there are exactly 5 tiles

Definition of an n-Mosaic An n-mosaic is an matrix of tiles, with rows and

Definition of an n-Mosaic An n-mosaic is an matrix of tiles, with rows and columns indexed An example of a 4 -mosaic

Tile Connection Points A connection point of a tile is a midpoint of an

Tile Connection Points A connection point of a tile is a midpoint of an edge which is also the endpoint of a curve drawn on a tile. For example, 0 Connection Points 2 Connection Points 4 Connection Points

Contiguous Tiles Two tiles in a mosaic are said to be contiguous if they

Contiguous Tiles Two tiles in a mosaic are said to be contiguous if they lie immediately next to each other in either the same row or the same column. Contiguous Not Contiguous

Suitably Connected Tiles A tile in a mosaic is said to be Suitably Connected

Suitably Connected Tiles A tile in a mosaic is said to be Suitably Connected if all its connection points touch the connection points of contiguous tiles. For example, Suitably Connected Not Suitably Connected

Knot Mosaics A knot mosaic is a mosaic with all tiles suitably connected. For

Knot Mosaics A knot mosaic is a mosaic with all tiles suitably connected. For example, Non-Knot 4 -Mosaic

Figure Eight Knot 5 -Mosaic

Figure Eight Knot 5 -Mosaic

Hopf Link 4 -Mosaic

Hopf Link 4 -Mosaic

Borromean Rings 6 -Mosaic

Borromean Rings 6 -Mosaic

Notation Set of n-mosaics Subset of knot n-mosaics

Notation Set of n-mosaics Subset of knot n-mosaics

Cut & Paste Moves

Cut & Paste Moves

Terminology: k-Submosaics Def. A k-submosaic of an n-mosaic M is k x k submatrix

Terminology: k-Submosaics Def. A k-submosaic of an n-mosaic M is k x k submatrix of M

Terminology: k-Submosaic Moves Def. A k-submosaic move on a n-mosaic M is a mosaic

Terminology: k-Submosaic Moves Def. A k-submosaic move on a n-mosaic M is a mosaic move that replaces one ksubmosaic in M by another k-submosaic.

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1

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Two different ways of thinking of k-Submosaic Moves 1) Cut & paste operations on

Two different ways of thinking of k-Submosaic Moves 1) Cut & paste operations on n-mosaics 2) A permutation acting on the finite set of knot n-mosaics. Observation: But all such moves are involutions, Hence, all such moves are products of disjoint transpositions.

Planar Isotopy Moves

Planar Isotopy Moves

Non-Determistic Tiles We use the following tile symbols to denote one of two possible

Non-Determistic Tiles We use the following tile symbols to denote one of two possible tiles: For example, the tile or denotes either

11 Planar Isotopy (PI) Moves on Mosaics

11 Planar Isotopy (PI) Moves on Mosaics

Planar Isotopy (PI) Moves on Mosaics It is understood that each of the above

Planar Isotopy (PI) Moves on Mosaics It is understood that each of the above moves depicts all moves obtained by rotating the sub-mosaics by 0, 90, 180, or 270 degrees. For example, represents each of the following 4 moves:

Planar Isotopy (PI) Moves on Mosaics Each of the PI 2 -submosaic moves represents

Planar Isotopy (PI) Moves on Mosaics Each of the PI 2 -submosaic moves represents any one of the (n-k+1)2 possible moves on an n-mosaic

Planar Isotopy (PI) Moves on Mosaics Each PI move acts as a local transformation

Planar Isotopy (PI) Moves on Mosaics Each PI move acts as a local transformation on an n-mosaic whenever its conditions are met. If its conditions are not met, it acts as the identity transformation. Ergo, each PI move is a permutation of the set of all knot n-mosaics In fact, each PI move, as a permutation, is a product of disjoint transpositions.

Reidemeister Moves

Reidemeister Moves

Reidemeister (R) Moves on Mosaics Reidemeister 1 Moves Reidemeister 2 Moves

Reidemeister (R) Moves on Mosaics Reidemeister 1 Moves Reidemeister 2 Moves

More Non-Deterministic Tiles We also use the following tile symbols to denote one of

More Non-Deterministic Tiles We also use the following tile symbols to denote one of two possible tiles: For example, the tile or denotes either

Synchronized Non-Determistic Tiles Nondeterministic tiles labeled by the same letter are synchronized:

Synchronized Non-Determistic Tiles Nondeterministic tiles labeled by the same letter are synchronized:

Reidemeister 3 (R 3) Moves on Mosaics

Reidemeister 3 (R 3) Moves on Mosaics

Reidemeister (R) Moves on Mosaics Just like each PI move, each R move is

Reidemeister (R) Moves on Mosaics Just like each PI move, each R move is a permutation of the set of all knot n-mosaics In fact, each R move, as a permutation, is a product of disjoint transpositions.

The Ambient Group We define the ambient isotopy group as the subgroup of the

The Ambient Group We define the ambient isotopy group as the subgroup of the group of all permutations of the set generated by the all PI moves and all Reidemeister moves.

Knot Type

Knot Type

The Mosaic Injection We define the mosaic injection

The Mosaic Injection We define the mosaic injection

Mosaic Knot Type Def. Two n-mosaics and same knot n-type, written are of the

Mosaic Knot Type Def. Two n-mosaics and same knot n-type, written are of the provided there exists an element of the ambient group that transforms into. Two n-mosaics and are of the same knot type if there exists a non-negative integer k such that

Quantum Mechanics in A Nutshell

Quantum Mechanics in A Nutshell

Q. M. in a Nutshell • The state of a quantum system is a

Q. M. in a Nutshell • The state of a quantum system is a vector (a. k. a. , a ket) in a Hilbert space , i. e. , • A quantum state at time changes into a state at time a unitary transformation , i. e. , where lies in the unitary group via

Q. M. in a Nutshell (Cont. ) • An observable operator, i. e. is

Q. M. in a Nutshell (Cont. ) • An observable operator, i. e. is a Hermitan • An important observable is the Hamiltonian , which is the Q. M. analog of the total classical energy of a physical system.

Q. M. in a Nutshell (Cont. ) • The time evolution of the state

Q. M. in a Nutshell (Cont. ) • The time evolution of the state a quantum system is determined by Schoedinger’s equation: • Given two quantum systems & of respectively in states & , the state of the bipartite system is

 • Measurement: Observable In Q. M. in a Nutshell (Cont. ) Eigenvalue Macro.

• Measurement: Observable In Q. M. in a Nutshell (Cont. ) Eigenvalue Macro. World Physical Reality Out Philosopher Turf Black. Box Q. Sys. State where Quantum World Q. Sys. State Spectral Decomposition

Q. M. in a Nutshell (Cont. ) • Q. M. is not well understood,

Q. M. in a Nutshell (Cont. ) • Q. M. is not well understood, i. e. , Measurement Physicists discussing Q. M.

Quantum Knots & Quantum Knot Systems

Quantum Knots & Quantum Knot Systems

The Hilbert Space of n-mosaics Let be the 11 dimensional Hilbert space with orthonormal

The Hilbert Space of n-mosaics Let be the 11 dimensional Hilbert space with orthonormal basis labeled by the tiles We define the Hilbert space as of n-mosaics This is the Hilbert space with induced orthonormal basis

The Hilbert Space of n-mosaics We identify each basis ket a ket labeled by

The Hilbert Space of n-mosaics We identify each basis ket a ket labeled by an n-mosaic row major order. with using For example, in the 3 -mosaic Hilbert space , the basis ket is identified with the 3 -mosaic labeled ket

Identification via Row Major Order Let be the 11 dimensional Hilbert space with orthonormal

Identification via Row Major Order Let be the 11 dimensional Hilbert space with orthonormal basis labeled by the tiles Construct Mosaic Space Select Basis Element Row Major Order

The Hilbert Space of Quantum Knots The Hilbert space of quantum knots is defined

The Hilbert Space of Quantum Knots The Hilbert space of quantum knots is defined as the sub-Hilbert space of spanned by all orthonormal basis elements labeled by knot n-mosaics.

An Example of a Quantum Knot

An Example of a Quantum Knot

The Ambient Group as a Unitary Group We identify each element linear transformation defined

The Ambient Group as a Unitary Group We identify each element linear transformation defined by with the Since each element is a permutation, it is a linear transformation that simply permutes basis elements. Hence, under this identification, the ambient group becomes a discrete group of unitary transfs on the Hilbert space.

An Example of the Group Action

An Example of the Group Action

The Quantum Knot System Def. A quantum knot system is a quantum system having

The Quantum Knot System Def. A quantum knot system is a quantum system having as its state space, and having the Ambient group as its set of accessible unitary transformations. The states of quantum system are quantum knots. The elements of the ambient group are quantum moves. Physically Implementable

The Quantum Knot System Physically Implementable Choosing an integer n is analogous to choosing

The Quantum Knot System Physically Implementable Choosing an integer n is analogous to choosing a length of rope. The longer the rope, the more knots that can be tied. The generators of the ambient group are the “knobs” one turns to spacially manipulate the quantum knot.

Quantum Knot Type Def. Two quantum knots and of the same knot n-type, written

Quantum Knot Type Def. Two quantum knots and of the same knot n-type, written are provided there is an element s. t. They are of the same knot type, written provided there is an integer such that

Two Quantum Knots of the Same Knot Type

Two Quantum Knots of the Same Knot Type

Two Quantum Knots NOT of the Same Knot Type

Two Quantum Knots NOT of the Same Knot Type

Hamiltonians of the Generators of the Ambient Group

Hamiltonians of the Generators of the Ambient Group

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

Hamiltonians for Each generator is the product of disjoint transpositions, i. e. , Choose a permutation so that Hence, , where

Hamiltonians for Also, let , and note that For simplicity, we always choose the

Hamiltonians for Also, let , and note that For simplicity, we always choose the branch .

Hamiltonians for Using the Hamiltonian for the Reidemeister 2 move and the initial state

Hamiltonians for Using the Hamiltonian for the Reidemeister 2 move and the initial state we have that the solution to Schroedinger’s equation for time is

Observables which are Quantum Knot Invariants

Observables which are Quantum Knot Invariants

Observable Q. Knot Invariants Question. What do we mean by a physically observable knot

Observable Q. Knot Invariants Question. What do we mean by a physically observable knot invariant ? Let Then a quantum operator on the be a quantum knot system. observable is a Hermitian Hilbert space.

Observable Q. Knot Invariants Question. But which observables actually knot invariants ? are Def.

Observable Q. Knot Invariants Question. But which observables actually knot invariants ? are Def. An observable is an invariant of quantum knots provided for all

Observable Q. Knot Invariants Question. But how do we find quantum knot invariant observables

Observable Q. Knot Invariants Question. But how do we find quantum knot invariant observables ? Theorem. Let knot system, and let be a quantum be a decomposition of the representation into irreducible representations. Then, for each , the projection operator for the subspace is quantum knot observable.

Observable Q. Knot Invariants Theorem. Let be a quantum knot system, and let be

Observable Q. Knot Invariants Theorem. Let be a quantum knot system, and let be an observable be the stabilizer on. Let subgroup for , i. e. , Then the observable is a quantum knot invariant, where the above sum is over a complete set of coset representatives of in.

Observable Q. Knot Invariants The following is another example of a quantum knot invariant

Observable Q. Knot Invariants The following is another example of a quantum knot invariant observable: where, for example, is knot genus of the knot, or, crossing number. Thus, the energy levels are the distinct values of the invariant.

UMBC Quantum Knots Research Lab Sponsored by the L-O-O-P Fund

UMBC Quantum Knots Research Lab Sponsored by the L-O-O-P Fund

Gromit’s Quantum Computer

Gromit’s Quantum Computer

? ? ? • Quantum Knot Kit

? ? ? • Quantum Knot Kit

Weird !!!

Weird !!!