Physical Knots and Particle Topology L. H. Kauffman, UIC www.math.uic.edu/~kauffman

( including joint work with Sam Lomonaco, Sundance Bilson-Thompson, Jonathon Hackett ) 19-th Century Sir William Thompson (Lord Kelvin) Theory of (Knotted)Vortex Atoms Lord Kelvin’s Vortex Atoms Leadbetter - Besant Atom -- Inspired by Lord Kelvin’s Vortex Theory of Atoms Babbitt’s Earlier Version

Knotted Vortices

Creation and Dynamics of Knotted Vortices

Dustin Kleckner1 & William T. M. Irvine1

1James Franck Institute, Department of Physics, The University of Chicago, Chicago, Illinois

60637, USA

While tying a shoelace into a knot is a relatively simple affair, tying a field, for example a magnetic field, into a knot is a different story: the entire space-filling field must be twisted everywhere to match the knot being tied at the core. Knots and links have been conjec- tured to play a fundamental role in many physical systems, including quantum and classical

fluids1–5, electromagnetism6–8, plasmas9–11, liquid crystals12,13 and others 14–16. Creating topo- logically linked fields in the laboratory, however, poses significant experimental challenges.

In particular, vortex loops, such as smoke rings, but tied into knots, are the elemental ex- citation in topological fluid mechanics2–4, 17–20; despite being proposed over a century ago, they have never successfully been generated in experiment. Here we report the creation of isolated trefoil vortex knots and pairs of linked vortex rings in water, produced by a new method of patterning arbitrary vorticity in three dimensions. Using high speed tomography, we observe that both structures spontaneously untie/unlink themselves, resulting in pairs of unlinked rings. This change in topology is preceded by a stretching of the vortex loop(s) and a corresponding distortion constrained by energetics. This work establishes the existence of knotted vortices in real fluids, paving the way for the study of knotted excitations in turbulent

flows, and offering the first glimpse into universal aspects of topological flows.

1 FIG. 1. The creation of vortices with designed shape and topology. a, The conventional method for generating a vortex ring, in which a burst of fluid is forced through an orifice. b, A vortex ring in air visualized with smoke. c, A vortex ring in water traced by a line of ultra-fine gas bubbles, which show finer core details than smoke or dye. d-e, A vortex ring can alternatively be generated as the starting vortex of a suddenly accelerated, specially designed wing. For a wing with the trailing edge angled inward, the starting vortex moves in the opposite of the direction of wing motion f, The starting vortex is a result of conservation of circulation – the bound circulation around a wing is balanced by the counter-rotating starting vortex. g, A rendering of a wing tied into a knot, used to generate a knotted vortex, shown in h.

Knots andFig ureLinks1 - A knot diagram.

I

II

extra structure at the nodes (indicating how the curve of the knot passes over or under itself by standard pictorial conIIIventions).

Figure 1 - A knot diagram. Figure 2 - The Reidemeister Moves.

That is, two knots are regarded as equivalent if one embedding can be obtained I from the other through a continuous family of embeddings of circles in three- s s space. A link is an embedding of a disjoiint collection of circles, taken up to 1 2 ambient Braidisotopy. Fi Generatorsgure 1 illustrates a diagramm for a knot. The diagram is Hopf Link regarded both as a schematic picture of the knot, and as a plane graph with

5 -1 II s s 3 1

Trefoil Knot -1 = s s = 1 III 1 1

s s s s s s = 2 2 Figure 2 -=The Reidemeister Mo1ves. 2 1 1 That is, two knots are regarded as equivalent if one embedding can be obtained from the other through a continuous family of embeddings of circles in three- space. A link is an embedding of a disjoiint collection of circles, taken up to ambient isotopy. Figure 1 illu=strates a diagramm forsa knost. The d=iagrsam is s Figure Eight Knot regarded both as a schematic picture of the knot, and1as a pla3ne graph wi3th 1 Figure 4 - Closing Braids to form knots and links. 5 Figure 3 - Braid Generators.

Ambient isotopy is mathematically the same as the equivalence relation generated on diagrams by the Reidemeister moves. These moves are illustrated in Figure 2. Each move is performed on a local part of the diagram that is topologically identical to the part of the diagram illustrated in this figure (these figures are representative examples of the types of Reidemeister moves) without changing the rest of the diagram. The Reidemeister moves are useful in doing combinatorial topology with knots and links, notaby in working out the behaviour of knot invariants. A knot invariant is a function defined from knots and links to some other mathematical object (such as groups or polynomials or numbers) such that equivalent diagrams are mapped to equivalent objects (isomorphic groups, identical polynomials, identical numbers).

6

b CL(b)

Figure 5 - Borromean Rings as a Braid Closure.

7 Is it Knotted? (See Rob Scharein’s Program: KnotPlot) Knotted DNA - Electron Micrograph, Protein Coated DNA Molecule Work of Cozzarelli, Stasiak and Spengler.

Text Spengler SJ, Stasiak A, Cozzarelli NR. The stereostructure of knots and catenanes produced by phage lambda integrative recombination: implications for mechanism and DNA structure. Cell. 1985 Aug; 42(1):325–334. rotate

42 Kauffman & Lambropoulou DNA Knotting and Recombination

~

K 0

~

K 1

~

K Tangle Model for 2

DNA ~

K Recombination: 3 C. Ernst and ~

K D.W. Sumners. 4

Figure 28 - Processive Recombination with S = [ 1/3]. ⇥ Lets see what the form of the processive recombination is for an arbitrary sequence of recombinations. We start with

O = [a1, a2, , ar 1, ar] · · · I = [b1, b2, , bs 1, bs]. · · · Then

K[n] = N(O + (I + [n])) = N([a1, a2, , ar 1, ar] + [n + b1, b2, , bs 1, bs]) · · · · · · Patterned Integrity

The knot is information independent of the substrate that carries it. Knot Sets dimension. After that invention, it turned out that the diagrams represented knotted and linked curves in space, a concept far beyond the ken of those original flatlanders.

Set theory is about an asymmetric relation called membership.

We write a S to say that a is a member of the set S. And we are loathe to allow a to belong to b, b to belong to a (although there is really no law against it). In this section we shall diagram the membership relKnotatio nSets as follows:

a b

a

a b

The entities a and b that are in the relation a b are diagrammed as segments of lines or curves, with the a -curve passing underneath the b -curve. Membership is represented by under-passage of curve segments. A curve or segment with no curves passing underneath it is the empty set.

{ }

{ }

{ { } } Louis H. Kauffman Reflexivity and Eigenform

Define a * a = a, b * b = b and c * c = c. ment of the algebra and (x * x) * z = x * z for is a topological curl. If you travel along the And define a * b = c = b * a, a * c = b = c * a all x and z in M. curl you can start as a member and find that and b * c = a = c * b. A special case of this last property would after a while you have become the container. In other words, each element combines be where x * x = x for all x in M. We shall see Further travel takes you back to being a with itself to produce itself, and any pair of this property come up in the knot theoretic member in an infinite round. In the topolog- distinct elements combine to produce the interpretations of the next section. ical realm, Ω does not have any associated remaining element that is different from Suppose that M is a reflexive magma. Does paradox. This section is intended as an intro- either of them. The reader can verify that TRI M satisfy the fixed point theorem? We find duction to the idea of topological eigenforms, a is indeed a magma. For example, that the answer is, yes. subject that we shall develop more fully else- where. a * (b * c) = a * (a ) = a Fixed Point Theorem for Reflexive Magmas. Set theory is about an asymmetric relation (a * b) * (a * c) = (c) * (b) = a. Let M be a reflexive magma. Let F: M → M be called membership. Note also that the multiplication in this a structure-preserving mapping of M to itself. We write a ε S to say that a is a member of magma is not associative: Then there exists an element b in M such that the set S. In this section we shall diagram the F(p) = p. membership relation as follows: a * (a * b) = a * c = b (a * a) * b = a * b = c. Proof. Let F: M → M be any structure-pre- a We will return to this magma in the next serving mapping of the magma M to itself. b section and see that TRI is intimately related This means that we assume that F(x * y) = a ε b to the simplest knot, the trefoil knot. F(x) * F(y) for all x and y in M. a Another example to think about is OM, Define G(x) = F(x * x) and regard G: M → the free magma generated by one element J. M. Is G structure preserving? We must com- Here we consider all possible expressions and pare G(x * y) = F((x * y) * (x * y)) = F(x * (y * This is knot-set notation. ways that b can combine with itself and with y)) with G(x) * G(y) = F(x * x) * F(y * y) = In this notation, if b goes once under a, we other elements generated from itself. F((x * x) * (y * y)). write a = {b}. If b goes twice under a, we write Remarkably, the free magma is an infinitely Since (x * x) * z = x * z for all x and z in M, a = {b, b}. This means that the “sets” are complex structure. For example, note the fol- we conclude that G(x * y) = G(x) * G(y) for all multi-sets, allowing more than one appear- lowing consequences of the distributive law x and y in M. ance of a member. For a deeper analysis of the (here using XY instead of X * Y): Thus G is structure preserving and hence knot-set structure see (Kauffman 1995). there is an element g of M such that G(x) = This knot-set notation allows us to have J(JJ) = ((JJ)(JJ)) g * x for all x in M. Therefore we have g * x = sets that are members of themselves, = ((JJ)J)((JJ)J) F(x * x), whence g * g = F(g * g). For p = g * g, = (((JJ)J)(JJ))(((JJ)J)J)). we have p = F(p). This completes the proof. Ω In the free magma an infinite structure is Q.E.D. generated from one element and all its pat- This analysis shows that the concept of a terns of self-interaction. magma is very close to our notion of a reflex- Ω = {Ω} Suppose further that we assume that every ive domain. The examples of magmas related Ω ε Ω structure-preserving mapping of the magma to knot theory, given in the previous section, M is represented by an element of the magma show that magmas are not just abstract struc- and sets can be members of each other. M. This will place us in the position of creat- tures, but are related directly to the properties ing from the magma something like a reflex- of space and topology in the worlds of com- a ive domain. munication and perception in which we live. In the next section we shall see that mag- mas arise very naturally in the topology of a = {b} b knots and links in three-dimensional space. 7. Knot sets, topological b = {a} This is an excellent way to think about them, and it provides a way to think about reflexiv- eigenforms and the left- Here a mutual relationship of a and b is ity in terms of topology. Here we take an distributive magma diagrammed as a topological linking. abstract point of view and see when the struc- ture-preserving nature of elements of a We shall use knot and link diagrams to repre- a magma leads to the analog of a reflexive sent sets. More about this point of view can be a = {b, b} domain. found in the author’s paper “Knot Logic” b = {c, c} I shall call a magma M reflexive if it has the (Kauffman 1995). In this notation the b c = {a, a} property that every structure-preserving eigenset Ω satisfying the equation c mapping of the algebra is realized by an ele- Ω = {Ω}

130 Constructivist Foundations Rx = ~xx RR = ~RR

Russell Paradox (K)not.

A A does not belongs to A. belong to A. In the diagram above, we indicate two sets. The first (looking like the

mark) is the empty set. The second, consisting of a mark crossing

over another mark, is the set whose only member is the empty set.

We can continue this construction, building again the von Neumann

construction of the natural numbers in this notation:

{} dimension. After that invention, it turned out that the diagrams represented knotted and linked curves in space, a concept far beyond the ken of those original flatlanders. {{}}

Set theory is about an asymmetric relation called membership.

We write a S to say that a is a member of the set S. And we are loathe to allow a to belong to b, b to belong to a (altho{u g{h} th{e{re} }is } really no law against it). In this section we shall diagram the membership relation as follows:

Knot Sets { {} {{}} {{} {{}}} }

a b Crossing This notation allows us to also have sets that are members of t h e m s e l vase s , Relationship a

a b a Self-

The entities a and b that are in the relation a b are diagrammedMembership as segments of lines or curves, with the a -curve passing underneath the a a b -curve. Membership is represented by under-passage of curve segments. A curve or segment with no curves passing underneath it is the empty set. a = {a}

and sets can be members of each other. a Mutuality { } b

a={b}

b={a} { }

Mutuality is diagrammed as topological linking. This leads the question beyond flatland: Is there a topological interpretation for this way of looking at set-membership? { { } }

Consider the following example, modified from the previous one.

a = {}

a b = {a,a} b

topological

equivalence

a={}

a b={}

b

The link consisting of a and b in this example is not topologically linked. The two components slide over one another and come apart.

The set a remains empty, but the set b changes from b = {a,a} to empty. This example suggests the following interpretation. a

b a={b}

b={a}

Mutuality is diagrammed as topological linking. This leads the question beyond flatland: Is there a topological interpretation for this way of looking at set-membership?

Consider the following example, modified from the previous one.

a = {} a b = {a,a} Knot Sets are b “Fermionic”. Identical elements topological cancel in pairs. equivalence

a={} (There is no problem with a b={} invariance under third Reidemeister move.) b

The link consisting of a and b in this example is not topologically linked. The two components slide over one another and come apart. The set a remains empty, but the set b changes from b = {a,a} to empty. This example suggests the following interpretation. Borromean Rings

Green surrounds Red. Red surrounds Blue. Blue surrounds Green. Quantum Mechanics in a Nutshell 0. A state of a physical system corresponds to a unit vector |S> in a complex vector space. U 1. (measurement free) Physical processes are modeled by unitary transformations applied to the state vector: |S> -----> U|S>

2. If |S> = z1|e1> + z2|e2> + ... + zn|en> in a measurement basis {|e1>,|e2>,...,|en>}, then measurement of |S> yields |ei> with probability |zi|^2. An Entangled State LPSC-03-09 arXiv/nucl-th/0305076 Borromean binding∗

Jean-Marc Richard Laboratoire de Physique Subatomique et Cosmologie Universite´ Joseph Fourier–CNRS-IN2P3 53, avenue des Martyrs, 38026 Grenoble, France Abstract A review is first presented of the Hall–Post inequalities relating N-body to (N −1)-body energies of quantum bound states. These inequalities are then applied to delimit, in the space of coupling constants, the domain of Borromean binding where a composite system is bound while smaller subsystems are unbound.

I. INTRODUCTION which are interlaced in a subtle topological way (see Fig. 1) such that if any one of them is removed, the two other become There are many examples, at various scales, of compos- unlocked. The adjective Borromean is nowadays broadly ac- ite systems at the edge between binding and non-binding. cepted in the field of quantum few-body systems. In nuclear physics, a proton–proton or neutron–neutron pair Borromean binding is intimately related to two other fas- misses binding by a small margin, while a proton and a neu- cinating properties of few-body quantum systems. The Efi- mov effect [6] indicates that when the two-body energy van- tron form73.2a rather weakly bound deuteron. The existence of a near-threshold state can induce dramatic consequences, for in- ishes (e.g., by tuning the strength of the potential), a myriad stance on fusion probabilities [1]. A pair of charmed mesons of weakly-bound states show up in the three-body spectrum. is presumably near the border separating stability from spon- This implies that the three-body ground-state already exists taneous dissociation [2]. Atoms such as 4He were for a long at this point. Slightly above the onset of two-body binding, time believed to be unable to merge into a molecule. Recent the ratio E2/E3 of two-body to three-body binding energies Figure 1. The Hopf link. 4 is very small. By rescaling, one can reach a situation with a fi- studies indicates a tiny binding of the order of 1 mK for He2. However, if one replaces one of the 4He by an atom contain- nite 2-body energy, and a 3-body energy that becomes infinite and gives a specific example3of a unitary braiding operator3 , showing4 that it does entangle quantum when the range of the potential is made shorter and shorter: ing the ligstates.hteSectionr iso3tendsopwithe aHlisteof,problems.then tSectionhe 4Hdiscussese Hthee linkis inuvnariantsboassociatedund. For with the braiding operator3 R introduced4 in the previous section. Section 5 is a discussion of the this is the Thomas collapse [7]. a recent restructureviewof entanglementon HeinNrelationHetoMmeasurement.systemSections, s6eisea,n eintroduction.g., Retofthes.virtual[3, 4]. braid group, an extension of the classical braid group by the symmetric group. We contend This review is organised as follows. In Sec. II, the Hall-Post An intrthatigunitaryuingrepresentationsquestioofn theisvirtualwhebraidthegroupr itproisvideeaa goodsiercontetoxtbandinlanguaged thrforee or quantum computing. Section 7 is a discussion of ideas and concepts that have arisen in the course inequalities are briefly recalled. They are applied in Sec. III to more comofpthisonresearch.entsAnthappendixan todescribesforma unitarya mrepresentationere twoof-btheothree-stranddy boubraidndgroupstate. and its relationship with the Jones polynomial. This representation is presented for contrast since constraint the domain of coupling constants leading to Bor- An answeitrcanisbepusedrotovdetectidehighlyd bynon-trithevialstopologicaltudy ostates,f habultoit doesnunotcleinviolv, we anhyiquantumch con- tain peripentanglement.heral neutrons. Consider for instance the 6He nu- romean binding for bosons interacting through short-range forces. The difficulties arising in the case of fermions are de- cleus. It 2.isThesttemptationable agofatangledinst astatesny dissociation, while the lighter 5 4 scribed in Sec. IV. Borromean binding with Coulomb forces He sponIttaisnquiteeotemptingusly dtoemakcaeyansanalogyintobetweena netopologicalutron entanglementand a inHthee.formInof the linked loops in three-dimensional space and the entanglement of quantum states. A topological is the subject of Sec. V, before the conclusions. (reasonabentanglementle) appisroa non-localximastructuraltion wfeatureherofea topologicalthe strsystem.uctuArquantume of tentanglementhe core is is a non-local structural feature of a quantum system. Take the case of the Hopf link of linking neglectednumber, thiones m(seeefigureans1).thInathist tfigurehe we(αsho,wna simple, n)linkthofrtweeo components-body andsystatesteitsm is inequivalence to the disjoint union of two unlinked loops. The analogy that one wishes to draw bound, wishwithilea stateneiofththeeformr (α, n) nor (n, n) have a discrete spec- trum. ψ = ( 01 10 )/√2 Quantum| ⟩ − | ⟩ Entanglement and II. HALL–POST INEQUALITIES which is quantum entangled. That is, this state is not of the form ψ1 ψ2 H H where This prHoispaecomplerty xovectorf 3space-boofddimensiony bindtwio.nCuttingg wiatcomponenthout 2⊗of-thebo∈linkdy⊗remobviens itsding topological entanglement.TopologicalObserving the stateEntanglementremoves its quantum entanglement in this case. was astutelyAnneaxamplemedof AraBvindorr[1o] makmeesathenpossibility[5], aofftsucheratconnectionhe Boervrenomoremetantalizing.an rings, A number of inequalities can be written down for binding Aravind compares the Borromean rings (see figure 2) and the GHZ state arXiv:nucl-th/0305076 v1 26 May 2003 ψ = ( β β β α α α )/√2. energies in quantum mechanics if one splits the Hamiltonian | ⟩ | 1⟩| 2⟩| 3⟩ − | 1⟩| 2⟩| 3⟩ The Borromean rings are a three-component link with the property that the triplet of into pieces (each piece being hermitian). Thus, for example, components is(|000>indeed topologically - |111>)/Sqrt(2)linked, but the removal of any single component leaves a pair of unlinked rings. Thus, the Borromean rings are of independent intellectual interest as

New Journal of Physics 4 (2002) 73.1–73.18 (http://www.njp.org/) H = A + B + · · · ⇒ E(H) ≥ E(A) + E(B) + · · · , (1)

in an obvious notation where E(H) is the ground-state energy of H. Saturation is obtained if A, B, etc., reach their mini- mum simultaneously. If, for instance, H = p2 − 1/r + r2/2 describes the motion of a particle feeling both a Coulomb and an harmonic potential, then E(H) ≥ (−1/2) + (3/2), cor- Is the Aravind analogy only responding to an equal share of the kinetic energy. A slight 2 FIG. 1superficial?!: Borromean rings improvement is obtained by writing H = αp − 1/r + (1 − α)p2 + r2/2 , and optimising α. ! " The reasoning can be applied to obtain a lower bound on ! " 3-body energies in terms of 2-body energies. This has been ∗Dedicated to my colleague and friend Vladimir Belyaev at the occasion of discovered independently by several authors working on the his 70th birthday stability of matter [8] or baryon spectroscopy in simple quark

1 Compare |000>+|111> and |100>+|010>+|001>.

In the second case, observation in a given tensor factor yields an entangled state with 50-50 probability.

In this way, we can make a case for quantum knots. You can imagine a topological state that is a superposition of multiple link types. Quantum Knots 3. WHAT IS A QUANTUM KNOT?

★✥ ✘✘ ⑦ ✟✟ ⑦⑦ + ✏①❛ ⑦⑦ ✏✏ ❛❛ ⑦ ✧☎ ✦ ☎☎ ❍ ✘✘✿ ✓✘❍✘ ✘✘✓✘

❄ ✘✘✘ ✘✘ ✘✘ ✦✦✦ ❧ ✏ ❧ ✏✏ ✏✏ ✏✏ ✏ ❩ ◗◗ ❩ ◗ ❩ ◗ ❩ ◗ ❩ ◗ ◗ ❩ ◗ ❩ ◗ ObservingFigure 2 - Obs earv iQuantumng a Quantum KKnotnot

Definition. A quantum knot is a linear superposition of classical knots.

Figure 2 illustrates the notion that a quantum knot is an enigma of possible knots that resolves into particular topological structures when it is observed (measured).

For example, we can let K stand for the collection of all knots, choosing one representative from each equivalence class. This is a denumerable collection and we can form the formal infinite superposition of each of these knots with some appropriate amplitude ρ(K)eiθ(K) for each knot K K, with ρ(K) a non-negative real number. ∈ iθ(K) Q = ΣK∈Kρ(K)e K . | ⟩ We assume that 2 ΣK∈Kρ(K) = 1. Q is the form of the most general quantum knot. Any particular quantum knot is obtained by specializing the associated amplitudes for the individual knots. A measurement of Q will yield the state K with probability ρ(K)2. | ⟩

An example of a more restricted quantum knot can be obtained from a flat diagram such that there are two choices for over and under crossing at each node of the diagram. Then we can make 2N knot diagrams from the flat diagram and we can sum over representatives for the different classes of knots that can be made from the given flat diagram. In this way, you can think of the flat diagram as representing a quantum knot whose potential observed knots correspond to ways to resolve the crossings of the diagram. Or you could just superimpose a few random knots.

3

3. WHAT IS A QUANTUM KNOT?

★✥ ✘✘ ⑦ ✟✟ ⑦⑦ + ✏①❛ ⑦⑦ ✏✏ ❛❛ ⑦ ✧☎ ✦ ☎☎ ❍ ✘✘✿ ✓✘❍✘ ✘✘✓✘

We need Quantum3. WHAT IS A Q UKnots!ANTUM KNO❄T? ✘✘ # ✘ ⇥ ✘ ⇥ ✘⌃ ✦✦ + ✘✘⌃x ⇤⇤⇤↵↵ ✦ a|K> + b|K’> ⌅ ↵ ❧ "! ⌅⌅ ✏✏ ❧ ⇥⇥✏⇥⇧ ⇥⇥✏ ✏⇥⇥✏ ✏✏ ◗ ⌦ ❩❩ ◗ ⇥⇥⇥ ⇥⇥ ⇥⇥ ◗ ❩ ⇤ ⇤⇤ ◗ ❩ ⇤⇤ K: probability |a|^2 ⇤⇤ ◗ ❩ ⇤ ⌥⌥ ◗ ⌥ ❩ ⌥ ◗ ⌥ ❩ ⌥ ◗ K’:probability |b|^2 ⌥ ⌥ ◗ ⌥

Figure 2 - ObsKerving a Quantum Knot K’ Figure 2 - Observing a Quantum Knot DefinitionObserving. A quantum knot is a alin earQuantumsuperposition of classi calKnotknots. Figure 2 illustrates the notion that a quantum knot is an enigma of possible knots that resolves into particular topological structures when it is observed (measured). Definition. A quantum knot is a linear superposition of classical knots. For example, we can let K stand for the collection of all knots, choosing one representative from each equivalence class. This is a denumerable collection and we can form the formal infinite superposition of each of (or a linear superpositionthese knots with some approp ofriate amprepresentativeslitude (K)ei(K) for each knot K forK, wi thknot(K) a non -ntypes.)egative real number. i(K) Figure 2 illustrates the notion that Qa=quK aKn(tKu)em kKn. ot is an enigma of possible knots that resolves into particular | ⇥ topological structWue rassuesmewthathen it is observed (measured). 2 K K(K) = 1. Q is the form of the most general quantum knot. Any particular quantum knot is obtained by specializing the associated amplitudes for the individual knots. A measurement of Q will yield the state K with probability For example,(Kw)2e. can let K stand for the collection of a|ll⇥ knots, choosing one representative from each

equivalence class. TAnhexiamps ilesofaa mordernesturimctedeqruanatbumleknotcocanllbeecobttiaionedn fraomnadflatwdiagre amcasunch fthoatrtmhere tarhe tewo formal infinite superposition of each of choices for over and under crossing at each node of the diagram. Then wieθcan(Kmak) e 2N knot diagrams from the these knots with flatsodmiagream aanpd wpercanopsurmiaovterereparesenmtpatliviestufordthee dρi⇥(erKent cl)assese of knotsftohatr caneabce hmadke fnromottheK K, with ρ(K) a non-negative real given flat diagram. In this way, you can think of the flat diagram as representing a quantum knot whose potential ∈ number. observed knots correspond to ways to resolve the crossings of the diagram. Or you could just superimpose a few random knots. iθ(K) Q = ΣK∈Kρ(K)e K . | ⟩ We assume that 2 ΣK∈Kρ(K) = 1. Q is the form of the most general quantum knot. Any particular quantum knot is obtained by specializing the associated amplitudes for the individual knots. A measurement of Q will yield the state K with probability ρ(K)2. | ⟩

An example of a more restricted quantum knot can be obtained from a flat diagram such that there are two choices for over and under crossing at each node of the diagram. Then we can make 2N knot diagrams from the flat diagram and we can sum over representatives for the different classes of knots that can be made from the given flat diagram. In this way, you can think of the flat diagram as representing a quantum knot whose potential observed knots correspond to ways to resolve the crossings of the diagram. Or you could just superimpose a few random knots.

3 Quantum knots and mosaics

Here is yet another way of finding quantum knot invS.aJ.riants:Lomonaco, L. H. Kauffman Keywords Quantum knots Knots Knot theory Quantum computation Quantum I(n)nf Process· · · · Theorem 3 Let Q Quantum, algorithms(n) beQuantuma quantumvortices knot system, and let ! be an observable A · K (n) on theDOIHilbert10.1007/s11128-008-0076-7space Mathematics. LetSubjectSt (!Classification) be the(2000)stabilizerPrimary 81P68subgr57M25oup81P15for !, i.e., ! 57M27 Secondary" 20C35 · · · K ·

1 Introduction 1 St (!) U A(n) U!U − ! . The objective=of this paper∈is to set the foundation: for a research= program on quantum knots.1 # $ Then the observable For simplicity of exposition, we will throughout this paper frequently use the term “knot” to mean either a knot or a link.2 In part 1 of this paper, we create a formal system (K, A) consisting of 1 (1) A graded set K of symbol strings,Ucalled!Uknot−mosaics, and (2) A graded subgroup A, called the knot mosaic ambient group, of the group of all permutationsU ofAthe(n)/Sset of knott(!mosaics) K. We conjecture that∈ the%formal system (K, A) fully captures the entire structure of tame knot theory. is a quantumQuantumknot n-invariant,Three exampleswofherknotsknotemosaics are given beloandw: U!U 1mdenotesosaicsa sum over a U A(n)/St(!) − complete set of coset representatives for the∈stabilizer subgroup St (!) of the ambient & grwithoup A(n) . 1 PrSamoof The observable g (n) g!g− is obviously an quantum knot n-invariant, since Samuel J.∈ALomonaco Louis H. Kauffman g g!g 1 g 1 g!g 1 for all g (n). If we let St (!) ′ g A(n) − &′− g A(n) − ′ A Lomonaco∈ Each of=these knot mosaics∈ is a string made up of·the following∈11 symbols | | denote' the order of (St (!) , and if we let c1, c2, . . . , cp denote a complete set of & | | & p 1 1 coset representatives of the stabilizer subgroup St (!), then j 1 cj !cj− St(!) 1 = = | | g!g is alsocalleda mosaicquantumtiles. knot invariant. g A(n) − ∈ An example of an element in the mosaic ambient group A is the mosaic&Reidemeister ⊓% 1 move illustrated below: &We endEachthis section mosaicwith an example is ao ftensora quantum knot productinvariant: of Example 2 The following observelementaryable ! is an example tiles.of a quantum knot 4-invariant:

1 A PowerPoint presentation of this paper can be found at http://www.csee.umbc.edu/~lomonaco/Lectures. html. 2 For references on knot theory, see for example [4,10,13,20].

123

RemarkThis6 Fobservableor yet another approach isto aquantum quantumknot measurement, knotw einvariantrefer the reader to the©briefSdiscussionpringeronSquantumcience+Businessknot tomography foundMedia,in itemLLC(11) in 2008the conclu- sionforof this 4x4paper .tile space. Knots have characteristic

4 Conclusion: Openinvariantsquestions and futurin nxne directions tile space.

There are many possible open questions and future directions for research. We mention only aAbstractfew. In this paper, we give a precise and workable definition of a quantum knot (1) Wsystemhat is the exact, thestructurestatesof the ambientof whichgroup A(n)aandreitscalleddirect limitquantum knots. This definition can be viewed as a blueprint for theA constructionlim A(n). of an actual physical quantum system. Moreover, this = −→ Candefinitionone write down oanfexplicita quantumpresentation for knotA(n)? forsystemA? The fact thatistheintended to represent the “quantum embodi- ambient group A(n) is generated by involutions suggests that it may be a Coxeter group.ment”Is it a Cooxeterf a group?closed knotted physical piece of rope. A quantum knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular 123 spatial configuration of the knot tied in the rope. Associated with a quantum knot sys- tem is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a quantum knot can exhibit non-classical behavior, such as quantum superposition and quantum entanglement. This raises some interesting and puzzling questions about the relation between topological and quantum entanglement. The knot type of a quantum knot is simply the orbit of the quantum knot under the action of the ambient group. We investigate quantum observables which are invariants of quantum knot type. We also study the Hamiltonians associated with the generators of the ambient group, and briefly look at the quantum tunneling of overcrossings into undercrossings. A basic building block in this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory.

S. J. Lomonaco (B) University of Maryland Baltimore County (UMBC), Baltimore, MD 21250, USA e-mail: [email protected] URL: http://www.csee.umbc.edu/ lomonaco ∼ L. H. Kauffman University of Illinois at Chicago (UIC), Chicago, IL 60607-7045, USA e-mail: [email protected] URL: http://www.math.uic.edu/ kauffman ∼ 123 !"#$% &'("% )*++"#&% ,&-+% .-!"&% -&% +/"% 01#2*("% -2% *% #/-34'( Knot+#'*(-& +Woven*/"5#-&6 on the Surface of7 "a#" %Rhombic'0% 8-4% 9#*$ Triacontahedron:0% +/#""% 5'3"&0'-&*;% !'01*;'<*+'-&% -2% +/'0% ."*!'&=6

Courtesy>0% +/'0% +/#"" % of5'3 Robert"&0'-&*;% =" -Gray3"+#$% #and";*+"5 % +-% +/"% '&+#'&0'(% 2;*+ =-;5"&% Lynnclaire#*+'-% ="-3"+#$ % Dennis-2% +/"% +#"2-';% ,&-+? !"# $%&'()# *+%,()# -+./012.+'+3(# 2%4# 2# 5326+%7# -21/010( @/"% 0*3"% 5'*=#*30% +/*+% *#"% 10"5% 2-#% 5#*.'&=0% -2% ,&-+0% *&5% ;'&,0% (*& 4"% 10"5% +-% ';;10+#*+"% +/"% #";*+'-&0/')% -2% 3"34"#0/')% A2-#% *% &-&B 0+*&5*#5% !"#0'-&% -2% 0"+% +/"-#$C% 4$% 10'&=% +/"% (-&!"&+'-&% 0/-.& 4";-.6

D D ! 8 8

>&% +/'0% (-&!"&+'-&% "*(/% ;'&"% /*0% *% ;*4";% *&5% +/"% 4#-,"&% ;'&"% '0 '&+"#)#"+"5% *0% )*00'&=% 1&5"#&"*+/% +/"% 1&4#-,"&% ;'&"6% >2% D% '0% +/"% ;*4"; -2% +/"% -!"#(#-00'&=% ;'&"% *&5% 8% +/"% ;*4";% -2% +/"% 1&5"#(#-00'&=% ;'&"E +/"&% ."% 0*$% +/*+% FD% '0% *% 3"34"#% -2% 8F% *&5% .#'+"% % D% !% 86 sets (as sets) was discovered by John von Neumann as a method for creating representative multiplicities within set theory. Numbers are created from nothing but the act of forming a collection from previously created sets. This form of recursion does the collecting automatically.

It would be good to see how such structures are related to the interaction of polyhedra in three dimensional space. It is in fact easier to see how knots and links and weaves are related to circularities and self-reference since they are so close in formal structure.

III. The Trefoil Knot, the Pentagon and the Golden Ratio First we begin with an experiment that you can do with the trefoil knot and a strip of paper. Tie the strip into a trefoil and pull it gently tight and fold it so that you obtain a flat knot.

Figure 27 - Tightening the Ribbon Trefoil Knot to Form a Pentagon

Figure 26 - Tying a Ribbon into a Trefoil Knot Now it is intuitively clear that this pentagonal form of the trefoil

As you pull it tight with care a pentagon will appear! knot uses the least length of paper for a given width of paper (to make a flattened trefoil). At this writing, I do not have a proof of this statement, but stand by!

What we can do is to determine the length to width ratio of the strip of paper obtained from this construction, by cutting the knot exactly along the edges of the pentagon.

Figure 28 - Cut Strip to Form Just the Pentagon

Then unfold this strip. Figure 32 - The Internal Geometry of the Pentagon Gives Rise to the Golden Ratio

The small pentagon has its edge e and chord d labeled. We embed the small pentagon in the larger one and observe via a parallelogram and by similar triangles that d/e = (d+e)/d (similar triangles). Thus with F = d/e we have d/e = 1 + e/d whence F = 1 + 1/F. This is sufficient to show that F is the golden ratio. Note how the golden ratio appears here through the way that a pentagon embeds in a pentagon, a pentagonal self-reference.

We conclude that

L/W = 4(F+1)/Sqrt(3 - F^{2} + 2F) = 4(F+1)/Sqrt(2 + F)

Thus

L/W = 4(F+1)/Sqrt(2+F).

This shows how the golden ratio appears crucially in this fundamental parameter associated with the trefoil knot.

Numerically, we find that Is the Geometric Universe a Poincare Dodecahedral Space? The Poincare Dodecahedral space is obtained by identifying opposite sides of a dodedahedron with a twist. The resulting space, if you were inside it, would be something like the next slide. Whenever you crossed a pentagonal face, you would find yourself back in the Dodecahedron.

What Does This Have to do with Knot Theory? The dodecahedral Space M has Axes of Symmetry: five-fold, three-fold and two-fold.

The dodecahedral space M is the 5-fold cyclic branched covering of the three-sphere, branched along the trefoil knot.

M = Variety(x^2 + y^3 + z^5) Intersected with S^5 in C^3. So perhaps the trefoil knot is the key to the universe.

MODERN TIMES Quarks and Gluons - Gluon Flux http://www.physics.adelaide.edu.au/theory/ staff/leinweber/VisualQCD/Nobel/ index.html

arXiv:hep-th/0312133 v1 12 Dec 2003 ∗ A n w t t T G c η t s a m t W t t t h e h h i h i e t n ( o o i h l d e i p 1 e e a e e t d u . s n n e N h . o s 4 t t , e i s e e g t e r 4 r i b q : D w a a x d y e l e 0 S t u u a b s l m , x p w h ) i A T t e l e a 0 s p c n l b l i i a o b a o s r + p @ c p o h e c y b k i h l a n + s a r e e r n s u t i h s l e i e t e s l l r t m w e i u a a c o e a m g h t e a t v s o q r t d e y l e o m n p f e p u e u c t u n o y w G v s w d r o e l e p e s s t o e i e a c b l . c n a a i l e o y q u o t s i t a n a l l w B c l n r r i t u u h u s n l l n e u u o t o a g l . i e r a c o s s g a c r b m r e o f t e s e a o s x o s o s p u n 0 h a s a n t e f f f f b ( o r m r fi a s r u t 8 1 d a o i s s v r h n b T n r 5 u s t l a u a 0 e e a t a e e i r g 0 s c r t h b 3 h m i o t q l t 0 0 o i n e i e n p o p l u , e ) − m t l t e r i b i n . s g h i r o a e t A n t + c S o p e e s h r y e n t t r I , i p v p o e u t - n n n e K g M d s b i i r r o w s t 7 r t t d l t t t e e e o s e a u e r e a i 5 h g e t t d l a a c i d r e n c t e a n n e e i A o o l e a i b y d f a , n t n g g v c 1 m f m t c o a i i b e a i e - e l t o t 0 J n c g l h w 2 p r e u i i s l n t t a o s P o l c a , 8 s a . l 1 t 9 g e u h i r o n l n t t n m r e u d l r e h a a , y e s i o a d U e i b d d N e . t s o i e y fi s 2 c e l g a t i e f g s d t a t e n s d p t o c e i T n 0 l . l h r l q l a e i u l g n d p t s i 0 o e h c u o h t i m e e . n i s s f t C y i 6 r a s c b r r i y e s h o g a a a W l , a d i i s i e n r h t n l d ∗ n t t l l t i e a t l U e h e a g r i e o s s r n r t b t u e , h g t i i a e i p t v n s r e c a e t v d r l e η S i - i t n s s p e m e n r a i L h o u o m m e w a s l g a b ( p r s d s r c a s 1 c t a e i n e a t e r t m w o o 4 s a s n o c l d i d i f s o n a 1 i y b s i n i m e S n e s n l 0 i e s d e a w o t e i q ) n u b s U s , Y d t , r n d u i s e i c e b t o s a r t t t f a n c h o r r f i o n e h o l i e r b s m i t r g i v t o m s n e t p w v s e h , - a m h e p e r M e e e e r m v g i e w o o x s e i m n r e o n d t i Y n e s s w s r l r c u e p u i l y a a i t s o fi ? t g n i s l c t n d o i n l n t i s t y e u e t d g s e i i s d s h x o m o , l - i e i i k y m M m s n e p f b t t n t i o a l h e h t e p e a i r o a r r h l l n n a a a l c y t l l r i y e s s t t t - - - - - . . arXiv:hep-ph/0408025 v1 3 Aug 2004 a k s n f m n t f J t m F t l c c fl e t l fl k e fl c n i p r l ‡ † K a ∗ a o s a e o h h h h e E E e h n T a s v a l o n n e o u u u o l o . r t s r o r o i e t t l d e e e i n n e a l l c I I n o h t n l o o a x x x a e e M o e e s t p r l t f n e r d c i f o e o n c c t t l d - x o R e c c l e s t b t d g s s t r o i t t s i d t s c i t r o I f i l w r t t h t [ l r t u s . r r i s o s / w y e a u e , e t v l E u s 1 d a m I o t h i s h o o e e a e t i p h s i b s s t o l p . o e b ] o n S b K s n e n n t f e i k r n e S i r p a c e . s p s p o p o i n , r r e s n o r a i i e e k e i a n - s g u u p c p k U c c p l a i r F s l s l ) i a i h g s b l L s s k v l t s a n r t y y t l e o e o s y e R p o u F s n r h c e u o a a i , e u a i i a e e s e e L p i p c c s g e n w g c s o p c s n d o o c r d d ff c n a i o t . f r r d t i r o c e E T p g u i t s h a h o i o s t j n a d d o p o a a e t r d c n r e t k fi m o o n t t m a E i u e r i h i y r x m S h a u V r r h t h r o c r m l d S h h s n n e o e b n o e d e c c r m r e e . 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I m o : N d s i p r e w n l e W w g I s e l o h e v t r h u t e l k t n o h e b t o e h a a g i n e f T c a n k p h o n h e n a s b e t d p e s d e t t x n e ) b T t r l o q e n h a a fi a n m r i o . l a e t g t p m u g l e h o c h T , I s o g e o i q g d o o e n y r u r r m o t s g j o g i u h s r O t t n w fi h . r N s l t e v a o l a r e l s u . t t o e t o u o t t f g , i u o a e I h f t a s e b e u i , o n m a a t o x e f h c a t t s o e i m u s b p e y G h F a t s r r s g T n s w k t R s g n . p e b l e d t h l t . s t e e f e i g . a b e m h y e l u n s i n . a l d r d t e t s h s s l n i t c w d p o f i e u H e n s i R l t e w u n e g l c F t p i P u s c e t s l d ff h u e H t o s s i a p e i o s a l , o e o l t f h . n . b l a k t o i s t o l h t a o v a y m a n e e c k O i , T s e u a e r z s b e R t r f u i t h a e i n t a l a n n e c m b “ e . r r t t r a e n - y y e r n d p s a W c o t e e a i h n l n i d l e v p l D a b h t N . h e a n o n @ E s q t s b d f c o [ n h i o l e K n c a a a n c p s o a h i e r r 2 n o u l l e n l e t o t h g u u a s m e x w f v y l W t o c c l U V h e u u n e s P s r a r r e u , n o t h o n o c k l m fl l u N a h s a u r o a l s n s y w s c o , d r n m d l p r m , k . a h d t l o a y i r n e o I 3 C s e s e i t l u i i a n e n n m r y a e s h a n g i m y o c s O ( n r @ O i t o d s d l a b e p ] t , t i n r b e n ( t x S t fi f n i a l l t o h ) T t t m h s s n d a l e o e l t o i s o p f y s m u u d v h r e p r s c s y e , T : c s i e k R v U g o r l r o s u e g n e i i e p t a c v s r i i I i t o l a o m m t d b r n a g t t n e h s e u n s m a d r e e d d t i k b n s o S A o O h l m e e o S t . h o i a n r a d a i g m p o l a b s , c n a r y l s d n t i i r d n r m e s i a l m a r e t t t p i i n o v n l v s a p e a N v m h t n o n S o a r e e t A o c s a i . o e e P n e t l p l o c a e l o fi . e a n l d r t a f e h e l a g e r t e k l u g c t i i ( x l i r r u t s i o d u b f g t r i t i c t m f c f u r N d s o a t H i v l t a r n f i e n g n h a T m o e h a n o o , b u d i i n a p r l t e i c , s o 0 c o t t n 0 k e l i t n i o y e l e n e o n h n t i d s b t D o a ( S Y a s A m n i s e o k e h n ( s t b s l l d i n b g e . V t s d a u k 9 ) i i , s 9 i d o e n a l h s l i e n s l n l e k c m h i n s s e h Q i c s e i o u 8 S e e e a p s c 8 n d o t n c r e e o - s . s a t v c , o o L t g r s t m p o n 0 d a t h fl 0 t t g s u r i l I l f fl h r t C t h p a h f a s n e a l e t f s y s s a ” o w ) o d o g I s s C ) t o B y o u , a e u , i p a t r r t o a c l n . e , , d D . t m i t N t t n i n t p m r p i ) l e r x t w t e x t s s C g o t s s h u h m r u A u h y a d o k J k t e o o s r r p e o d c i r u g h e t a K w e n i e c d n m e t c fl . m a o o t a n i e b e n n u m l o L t o o l l o e b g t o h w o d fi o i , a i c c s u i r n d n e o C e h y s o e S k k s r , l , f y y e e n c n n t u i t e a i i g c n s 2 t y x w s h l p n t a s , r n n i d i r o † fi a fi a i t i w e fi o 0 t t c t o i d , o n g d t n t l i c o e . h g c e o a r o o m e 5 t r 5 g g t n n 0 v a o w n x V h e e n i h o a o u n i o r i d e a u a i i l y i s 4 o i u t t f u s n . d l n n d d e f a e c c e e e c a t s t s r ------. , l , . , , l . , b , n r r d w n a a e d t l d i t i t s r n i t i i e . a o g h o k T r d n O n i i a i q o k i Q a i t k e b l t t t T c t a T t t b t w b s e t c ( n c p o S t e a R t m c e b h i n n n n n a h p y q i h r i r h i h u i e h o x n o a a d i n f r t f u n u n l r e h u e i t g g o o h h i i e d t C u u l t g e e t v l o q s q i p e ¯ e e a b r l c n e , a a s I d h t h c t y s a o o y a m c fl h i h o n n a o o t t d i e , o m m e r ) n m h i a o n o e m r t D e h e s t a s l n t t t o s h h ≪ c n r m t t t u s ) r U e fl r f k u h i u v g k . m a h r , n e h s m s p i a n t r r h l o o , e k t , e i a s a M o x u c i o e a s y e n x b n i h s e a f j w b s u i m n c fl fi a p n k f a r c e a t m a l s a n a t p d T o x b e o w , r b r f p i i l M c n t e a a i e i m p l e u e n t t h e n o e n v o d e e v l y k c p g o o r M r r m r e r o y - r o f r p a s e e u h r h g r l W p a a t x e n ¯ x o a 1 h g e e e r a s d , c e n e f a d s n e c e u e t r d y i u h k p , e i b v e r a f l n t i r e w r t 9 l a t d r c y a a f m m fl r t o i i o l M a f d r o s u s n m o y l o b a . fl o n s v h . c r p b m ( a e d t s e o 9 m s t k p s i a e i t p i t i u s l e p a i s u t t o a e n i τ l n o e c o s i u u p o i t q y i d t a e r o h 0 b t a l F r t l e d i o l K c o m y x a M e l b s t r g o r o p e u k m r a g t e c n x s u s a i d t o e g u s e w a r e n k e ¯ i l i c , r n i w r s e m l t n a s n t i d h o l n n b e t r y r l e c g d n r e fl a i n , n n e o t d c c s n . [ e e . r t y n h s s l , o a n d / N fi t t p l e e 4 l l t h e e / t u ) a o a o a fi l d e l d g a s d d e i o a e c e i a a D e u t p e p n m n a s h i o i e p x , g ] s h t . f a t n a a x l l l r c s T e l r d r c u a . o / l s , e r n n T e l b a c l o u n r 3 h g a s i u e i r p s a n N i s s w n a r s d e y r a a 3 r x o n l e s n t A t d a a d s h c 1 i l n r m a t h t e T m e t e i o r e p e r s h , t y e s 1 e r o a l s p e i n p o i y i t s n u y e i A q n e m v b e d e a n r i o t m s i l - l l c e l c m - r a r e n e h n n . p r i m z f l - b e o e s e b e i t c l a e k b ‡ u a d c p t o f e g d a n t s p i s m e l r r a t l a n c e e i a o . c fi i p o e w i a t n e e e o W r r r r k l a a e t [ e t e a e t c a t v f c a e r o e t n s t a r t h i t 4 l r n o s d y u r d e l r i h t o s n n e i n o m w W l o o r c y m i e s r , g i r e g n e s a s a s t r u o s ] , p h a n a l a s d i s o m l t e h o i d e n i q t t s w n . s y d z t n , e n h t c e . i c l n f c t k v n T e i h i l t s s u i n p r s t s t e l q a i o i a o i ¯ h h n c s u / y i o o o e t g i s a h b p n l l c e h e e t i n i i d M y o o p m o l N m w fl t n f t p l e g n y n t a e e n n c i fi l l n s e e a e o t t u e p n g n e s e T h t f k r c o h i u u p o a r i c r o t s u n g g n o , n i fi t n r n c o d i a Q Γ a s h e r n d n a a s d n e n e Q d e e i t y t 3 d c x n t p l i r h s t t l p t t s d g o r c n d k t a i i e g s i e i s , a ff [ t e c s i 7 t s o Q o t i s i h a C h h n e t fl r i a e v e o r r 9 e i s C u n v e y - s e s m t s a . h h o n c i h 2 n e a t t c s e g b e e e c e m i n e u n ( ( c ] p e t u o D C r c = n h h s t r 3 s c r D e g t n u t c n i n M t T r s h n s u y a o t a t m x o l t r t v b r s t d e c l t i h 5 p s e o u o a s D o s o i i d o l g h o o o l o g w o u e o t t k t o m ) e e c c s t t h w s y s , e s a i e [ l n n w l b m d C o o f i i w 1 t i a p h o n ff n c h o e r 7 a r n e e o . p i t e s u s e o e i n l t h w b n o i r a t U o p a e / s e h d s t h t p n t o n M . h ] s - c e x e m t fl o r a l n i e n e W m g o e . g u r p s n e z e r e t τ l e h m m a e m l l . e r e a t s S e ¯ a a t w ff u r s s c i a T o s a n a k c e e d s n s p e l o v o e g r ( s C b T v τ r t a h a A h r m c e t g ∗ x y e c e q c ) e a t r n a . h n h t e n n e i e l e h t o s r e e e e a e s a , y a c i a ) a k o w c e h h q u [ k r h l e v o , o r o t a o t p o f w s e 5 s s t i c t p l n s n s s p q a t o g e w r i ¯ e e a o a ( a u p e f f t t i t e n p e c s l o u o a ] o s a i i o ( h a e o t i p e g s n M h c c e e r r u r e f n s a n , e f l e l . t a e f l b i n h b e t s p l c e r o l k e t m i a m l e a y r u t o r u d l a n d e f e i i e t t s i u m e c n j e w n a n / r h T p o m r t s M e p f c n t s u n t a h n t ff e t e t o a m p d s o s t ¯ r h s i i e a x fi i r d y s a a a u o o h t e o c s n r c d h n e d r i o r e s a n e e fl o r e n a h g v a l t r g r t p i ) f a i s d w t e s e f c i e e n p t g l o t i n s n a u a t l s p o s t m , u e a e l e e l y p y p i r n t o e i l p r i s ( e y s i f e i e i p n g v l t s x e d r s n e r n p w o r ) c o t r - r o i t l l o x a a f n n e p e t w c o o d o b t a o e s p a r q h o o e l k r r a a l r f a f s n s p x i m ( i o e a g t d l e c l l u s g d t t e l r u o r o d d d l s h r p p e e v a o n i i i e s a d l e t t l t t u w i y e e s l i n u t g n n s c t t t a , a a . t c u u o e e b f e e e h i r h c r c m s o g [ [ s i b h s h h h u v t h t o a t t u c r x i g r o d n o c 8 s c 6 t c a c a e t l i i ) i i n n . i o o o a k e e e e e e e e s s s s t r s s s s ------f , ] , l ] . l l . . The next demonstration is made by Jason Cantarella, using his program “ridgerunner”. http://www.math.uga.edu/~cantarel/

In the next frame we show a Cantarella film, contracting the knot 9_{42}. This is the first chiral knot that is undetected from its mirror image by the Jones polynomial.

And here is the contraction of the trefoil knot.

Mobius Strip Particles

(published in Journal of Knot Theory and Its Ramifications)

The Non-Locality of Impossibility

Analog: Imaginary Object Quantum State Observation Selective Ignorance

SU(2) versus SO(3) and the Dirac String Trick Orientation Entanglement Relation Corresponds to SU(2) Symmetry

i2 =j2 =k2 = ijk = -1

-k-k

Quantum Hall Effect

Braiding Anyons

Λ Recoupling

Process Spaces

The process space with three input P’s and one output P has dimension two. It is a candidate for a unitary representation of the three strand braids.

P P P P P P

|0>: * |1>: P

P P

For this specialization we see that the matrix F becomes 1/⇤ ⇤/ 1/⇤ ⇤/ 1/⇤ ⇤/ F = = = /⇤2 T ⇤/2 /⇤2 ( 2/⇤2)⇤/2 /⇤2 1/⇤ ⇥ ⇥ ⇥ This version of F has square equal to the identity independent of the value of , so long as ⇤2 = ⇤ + 1.

The Final Adjustment. Our last version of F su⇥ers from a lack of symme- try. It is not a symmetric matrix, and hence not unitary. A final adjustment of the model gives this desired symmetry. Consider the result of replacing each trivalent vertex (with three 2-projector strands) by a multiple by a given quan- tity ⇥. Since the has two vertices, it will be multiplied by ⇥2. Similarly, the tetradhedron T will be multiplied by ⇥4. The ⇤ and the will be unchanged.

ONothtierce tphartopiterfotllioeswsofrfomthtehemsyomdelmetwirylol frtemhe daiiangruamnmchaatincgredeco.upThlingefonr-ew recoupling mulas of Figure 36 that the square of the recoupling matrix F is equal to the

matrix, after such an adjustment is made, becomes identity. That is, / /⇥2 1 0 2 1/⇥ ⇥1/⇤ ⇤1/⇥ ⇥/ = F = 2 2 2 2 2 2 = 0 1 ⇥ /⇥ ⇥T ⇥//⇤⇥ /1⇥/⇤ T⇥⇥/ ⇥ Notice that it follows from the symmetry of the diagrammaticrecoupling for- 2 2 3 mulas of FigureFo36r sythamt mtheetrsquary wee rofeq1/the⇥uir+reec1oupling/⇥ 1matr/ +ixTF⇥ is/equal to the 3 2 2 4 . /⇥ + T/(⇥) 1/⇥ + ⇥ T / ⇥ identity. That is, 2 2 2 Thus we need the relation ⇤/(⇥ ) = ⇥ /⇤ . 1 0 1/⇥ ⇥/ 1/⇥ ⇥/ = F 2 = 2 = 0 1We take /⇥2 T ⇥/21/⇥ +1//⇥⇥2 =T1⇥. /2 F⇥or this specialization we see that the matrix F⇥becomes ⇥ 2 ⇥ 3 This i1s/⇤equi⇤v/2alent to sa1/⇤ying th⇤a/t 2⇥ 3=1/⇤ ⇤⇤//. F = 12 /⇥ +2 1=/⇥ 2 1/2 +2 T ⇥2 /= 2 /⇤ T ⇤/ ⇥ /⇤ ( /⇤ )⇤/2 ⇥ /⇤ 1/⇤ ⇥ For this specializa3 tion we see that the matrix F ⇥beco2m=es21 +4⇥, . WiThistvhersiotn/h⇥oifsF ch+ahs Tosqiu/cea(re⇥eqoufa)l ⇥to 1thw/e ⇥eidenh+tiaty⇥vinedepTen/dent of⇥the value of 1/⇤ ⇤/ 1/⇤ ⇤/ 1/⇤ ⇤/ F ,=so long as ⇤2 = ⇤ + 1=. = Thus we need thaequrelada/r⇤tai2otinTc⇤eq/u2 ationwh/⇤2ose( so2/⇤lu2)2t⇤io/n2s are /⇤2 1/⇤ ⇥ ⇥ ⇥⇥ 3 ⇥ The Final Astdjate usand atblmank spenace tfor.theOunumarkedlastastte. Thvener⇤onsie hoa/s ntwo(omo⇥fdesFof su⇥er)s=from⇤a lack/o(f symme-⇤ ) = 1/ ⇤. This version oinfterFactionhoaf asboxsqwithuitaselrf:e equal to the2identity independen⇥t of the value of try. It is not a symmetric matrix, and hence⇥not=uni(t1ary. A fin5a)l/a2dju. stment , so long as ⇤1.2Ad=jacen⇤cy: +11/. ⇥ + 1/⇥ = 1. ± Henof the mceodeltghiveesand tnhiews desirsyed symmmetetry. rConsideric F theisresultgivofenreplacingby teachhe equation This is equivalenFtrTheivalentutrttFiohnaersaverlmAytexdjoi2n.r(withusNeste,gintg:mtwhthren.eatet.ekO2-prnuorwojelastctorthverastrsitoands)n of Fbysua⇥ermultiples from abylaackgivenof symquan-me- With this convention we take the adjacency interaction to yield a single box, 2 2 titytry. ⇥It. Sisinnceottaahndesythenmestimnhg ainettserarcttiwocn toomprvoaderutcertnioixtces,hi,ng:anidt wihenllcebenmotu⇥lutnipitl=aiedry.bAy ⇥fin.a1Sl iamdijulastrlym, enthte 2 4 ⇥ toetf trhaedhmedodroelngTiveswitllhbisedmesiurl⇥tediplsyi=edm=mbyet1⇥ry+. ThConsider⇥e ,⇤1an/thed⇤threesult wiof1ll /breplacinge un⇤chanegached. ⇤ ⇥⇤ frOtrivalenttohmer pFiroverpgertexutires(withe o3f 3tthrh.e eHenme o2-prdelceoje=wiFctorll r=emstraands)in uncbyhaangmultipleed. Thebynewa givenrecouquan-pling= We take the notational opportunity to denote nothing by an asterisk (*). The ⇥ 2 mtityatr⇥ix. ,Sainftceertsusyhnetactihcal raulhesnaforsaopdterwjuatinogstthvemaerstenertiskitces,areiThsusmithteaawistdere,iskllisbba ecosteand1m-inm2u/esltipli⇤ed by ⇥ . S1im/il⇤2arly, ⇥the ⇥⇤ ⇤ ⇥ a quadratic equation whfoor nosemark atsoall andlitucantbeieroasednor pslacedawherreveer4 it is conven⇥ient to = ⇥ + 1 = . tetradhedron Tdo so.wiThulsl be multiplied by ⇥ . The2 ⇤ and the will be unchanged. = 1. /⇤ ⇤/⇥ Other properties of the model wi ll remain unchanged. The new recoupling We shall make a recoupling theory⇥ba2sedo/n t⇤his p2article, but1it/is⇤worth whWeersheal⇤l notinogisowsme of spits phureleciyecombiangatolroiialzeplropdertiestenfiorst. ⇥Thtehrariteahmetticac oif o⇥seawhndere⇤ = 1/⇤. This gives the Fibonacci model. matrix, after sucombcinhing abonxes (astadn⇥djuing stfor =mactsenof d(tist1inictsionm) accoarddine,gto5bthecoese)/rulesm2es. has been studied and formalized in [52] and correlated with Boolean algebra For symmetryandwcleassicarleqlogiuc. iHerree within and next to are±ways to refer to t2he two Using Fisidges duelineartedesby the giv3en d7istinctioan.1Fn/rom⇤dthis poi3nt o⇤f8vi/ew,⇥,⇥therwe=are eh=av(1e+th⇥a5t)/t2h,e local braiding matrix for the model Furthermore, we know thtwo amodtes of relationship (adjacency an2d nest2ing) th2at arise a2t once in the2 presence of a distinction. ⇤⇥/(⇥/⇤) =Fibonacci⇥ 1//⇤⇤ ⇥. properties (the operator is idempotent and a self-attached strand yields a zero is given by the formul2abelow withevaluaAtion) and=give diagreamm3atic prioo/fs o5f th.ese properties. lWFeaoer vtsyaikmnemgettrhy eweorteqhuerire ca⇥ses=for the1exploration of the reader. We then take P P P P Model 2 3 ⇥ 2 = ⇥⇤ 2/. 2 ⇤/(⇥ ) = ⇥ /⇤ . 4 3i/5 1/⇥ 4i/5 from Figure 33. Hence = With this choice of ⇥ we have AA = e0 e 0 We take 2 R P= 2 8 = . ⇥ 2 = ⇥2 + 1⇥=3 . 2i/5 so that Fi⇤gure/2(5 ⇥- *Fibona)cci=P⇥art⇤icl=eInt/⇥era(ct⇤ion3/⇤.) =01/⇥⇤.A 0 e ⇥ = 0 = ⇥ 62 2 2 We shall now spHenWiecithceathtlhiisezecnhewoitceosyomtf mh⇥etewreicach=aFvseeis gwhAivenerbyeAtheequ=ation 2cos(6⇥/5) = (1 + ⇥5)/2. The simplest2 exam⇥ple3 of⇥a braid group representation arising from this ⇤/(⇥ 1/⇤) = ⇤1/⇥/(⇤ ⇤ ) = ⇤1/ ⇥⇤⇤. Forbidden (1/⇥)⇥ F = ⇥= = 1/⇥ = = 0 Note that ⇥ =11//⇥⇤==(11/.⇤+Th⇥ u5s)⇥/⇤2, ⇤ ⇥ tHenheoce trhye newis sytmhmeetrriepc F risesengiven btyathteieqonuatioonfthe three strand braid group generated by S1 = where ⇤ is the golden ratio and ⇤ = 1/⇤. This gives2the Fibonacci model. 1/⇤ 1/⇥⇤ ⇤ ⇥⇤ = =T 1 leaving the other cases for the explorati=on(of t1h/e)rea der⇥./W =e t 1/h⇥ en1=.take RUsinagnFidgurSes 237 =aFnd=F38,RweFhave(tRhatemthe eml=ocal bbraeridintg hmatrtix Ffor t=he mFodel = F .). The matrices S1 and S2 1/⇥⇤ 1/⇤ ⇥ ⇥⇤ ⇤ ⇥ is given by the formula below withA = e3i/5. aarned both unitary,Aa=nde3thi/ey5 generate a denFigure 2se8 - The 2su-Projectbor set of the unitary group U(2), where ⇤ is the golden ratio 4and ⇤ = 1/⇤.4Thi/5 is gives the FibInFioFigngureurae 2cci299, w- eFishbmoownatohcciedessenPelartce.icloef tahse 2T-Pemroperjectley-Lorieb recoupling model A 0 2 2e 0 for the Fibonacci pa2rticle. The Fibonaccie particle is, in this mathematical Using Figures 37 and 38, we have that the local braiding mNoatetthrmaitoxidnelFi,Temperleyfogiduenrert2ifi9edtwhewiheathvethamdeo2p-potedrodjectaelsinorglietselst rf.aLiebnAsd ntohteatrieaondfoer rcathneseeparfrticlome Figure TR == ( 1/8 )=( 2) 2i/25 inter/.act2io9n, s,thwi=ertehaaresot(lwidostbraasincd icontrerrespactoion2ndsinog)fttohteh2e-pmraorjectk2edopr(awirtitclhe,itaseldof,ttoedne1giv)ing/ supplying the fi0rstApart 3oi/f05 thee trastnrand sfo(or nothingr) comrresponadingtto ithoe unmnarkedsparnticle. eedA dark vertexed for quantum computing. ⇥ ⇥ a 2-projector, the other giving nothing. This is the pattern of self-iteraction is given by the formula below with A = e . indicates either an interaction point, or it may be used to indicate the the so that ofRepresentationthe Fibonacci particle.There is a thirdpossibilit y, ofdepicted in Figure 29, single stwhranerdeitswshoo2r-pthraonjectd foorstwinoteroradctintaoryprstordaunceds.a R4-pemroemjectberor.thWatetcoheseuldarreme ark at The simp2lest examp2le of4 a braid=gr(oup4ir/ep5 1r)esen(tatio2anll sh)oar/thraeniodusitexsetnp=,ressitghaotntfrhs3efo4or-pumrnodjecterlyotirnwighblli5rbaseckzer.et opoiflywneocmhioaolsecatlhcuelbartaiocnks.et polynomial A 0 e 0 vari⇥able A = e3/5. Rather than start there, we will assume that the 4-projector = A ABraid = Representations2cos(6⇥/5) = (1 +In FiguresFibonacci30, 531, )32,/33, 234 a.nd 35 we haModelve provided complete diagram- is forbidden and deduce (below) that the theory has to be at this root of unity. theory is the represenR =tation of the8 thr=ee strand braid2gir/o5 upma.tigc caenlculerationas tofedall ofbthye relSeva1nt sm=all nets and evaluations that are 0 A ⇥ T 0 1 e ⇥useful in the two-strand theory that is being used here. The reader may wish NoR antdeSt2 h=aFtRF (RememDenseber that inF =UnitaryF = F .). Thetomskipaditrectrilycesto FiguSre 316a andnFidgureS362b6wh6 ere we determine the form of the recoupling coecients for this theory. We will discuss the resulting algebra Note that are Th1bo/tehsium=niptlaest1ry., exaThnadmtuphleyse ogfenaerGroupsbartaeida dgrenousep surepbTsetresen=ofttahteiobunelonw.2ait/rai⇥siryng2gr,frouopmUt(h2i)s, sutheopprlyyiinsgththeerepfirstresenparttatoifotnhoefttrhaensfothrreemasttiroannsdnbeedraiedd gfororuqpugaennterumatedco7mb9pyuSt1i6n7 =g. 2 T 1 frR oanmd Swh2 =iFcRh=F i(tR(emfolemlob1wser/thi)amt Fm=edF⇥ia=/telF y=.)t. hThaet m1a.trices S1 and S2 are both unitary, and they generate a dense subset of the unitary group U(2), and supplying the first part of the transform79ations neededFfo2r q=uanItu.m computing. 2 2 2 T = ( 1/) ( 2) 2/ = ( 2) 2( 1)/ ⇥ This proves that we can sa79tisfy thismodel when ⇥ = = (1 + 5)/2. = ( 1)( 2)/ = 3 5. Note that 78 T = 2/⇥2, from which it follows immediately that F 2 = I. This proves that we can satisfy this model when ⇥ = = (1 + ⇥5)/2.

78 The Fibonacci Model yields a braid group representation that is universal for quantum computation. It is a braid group representation that is dense in the unitary groups.

The structure of this representation is also theoretically realized in the present models of the quantum Hall effect. arXiv:hep-ph/0503213v2 27 Oct 2006 P i o o a l t h e i c r t h a c d t u t ∗ c i T s t i i o f b d t n e n n n n e h h i h a r u y h E v e a p n n n n i e e a i e c a a h r i t n g c g ff s i i e i e l e g a T r v l d d l p e l m n t o n s d l e 1 l e f l t f t e r e e c t g r e e d u i u i y l c o t h S s 9 g i i h s . r o n g q e d l e i r t m c n t . h S d c l t r o o e 7 e M r a t e m e i t s u r a a a h o m g a e h o i o s n s p n 4 e T t R s a o n i I n a t l d S h n m e n p e f u e p , t r , n e s n d d e p w r g h i i a o t l t o p d a l c p d o c a s d k e b d a c e e e a h h h e a e r p n o i r h s H n e c o l x t s t n i a t i S n l m t y o r n e t n i b n s f e t o h , s fi h d d l s o i o t y m w t a e i a d r s d e r o s , y c s [ n i a n e e o d i r e u o i d s m o P r a t p y t 3 l r o m t s a t t t u t e c c a s n t o a r r v d u e h h w m a f n m e e . i a g A r , r n h r s M e p v t i o a r e n o p t w m d e o t b o p e e t n W i r o d d e n h s p i , s C e u e a r l c r o C d s o r 4 i s l s c t e s l v n c m a f t r o I p r r e r ’ v t “ e s s : e e S t u h t e d e e u o p l ] m M s o c e i . e e o e s s c h m c s r d r h e o g n r l s - o c s f e t o n l r o a r o i s e e h d n u n i f s e m s e w a e e p b a e d t t r t a h h m o o l s o a o p n n e o r o c e d f u q e a c f s y . e l h i v n e e d f r e r o c n l f d e h s I t c s j , l l a o m s t e i a u o r l p q o w s e e e e u r m d i d e N e f r c o u a - i a e i t n g n a o a m n p m ν r t a r i r a h c b u c s O m m c r s s h e a u c p r l s n n h a r e t i u g d e e t m p r i t i r u T h r e s t t a a e r k s o e e a s b o i e r @ n w t t e o o ( s e k r a t e h t r o - l i h p r a m e o b e u e n h i n S s R n r o e w C c i t e r n h e t p r s i p n s p o e k o o n w : c x d o g n s d f o c e w f j t t p o y i s e M s . g o s h o a e r - n i e s O s l s n e a h d w a n h d e , d f 1 , r e q m r i t i e o + y o “ n r o c h s n n s E c l n a n 2 e l e e l h s e s s ) t u s a o s s D u s t l h e p t o c t v t t f t e o a . s n t i u e n m n e h e i i , s q r u p h a n o 6 k f d m l f o l t c e n i e p i o + a n s m e e ( g y / v b q y o t a U c c e fi 0 r u ” s c h r s n i r d o a A u t o e s l m n r r s a 3 p u h k e n . i t . , s s t e h e e e h i f y t a a e e R t o . t n a f n C m b s a b h i d s r v u b s l d a d g e e a p w o l d a t e i d m b s t t v e u l w i , d e l c U fi e b - a i e n n d a - o g e . o r n . s f h T o i e t y b a s p o q , t w i e a c o p l k a c y l m t l l s i l u f o g e h b d n e i m d n v r e l l e a t c n u 1 o l m i t t f a n I T s t o n t [ c i i e l p n l l t y e i e w e q - n i 2 t h i o e o 2 n t y h e O b o m y g a t l r o t v f d t i t e s p a o e m t r h s o h h p . s p e u v ] e t d t a h e g i n w e e i h r , w o e 1 o P e x t ) a - n a s s N d ‘ - s h o e e e r i o a e a o r n k . e o a o 0 e e k h , s n r s e c r f , n n p n e e S s l s a e e s l s n a l a n t . r t l u − d i / n o e s i o i d , s n e d o p l e r D s e i t l b o l d t s u t w p e s e b U n t u n o p s e t n l a o u o u m i g e c b − a h i o a t h e m n e b f d u y m s e S u i e l d c u d a h o . t n r a i s r c r n o t i a s a f m n t c q g o h a a o x c t r b r n l u r p b h . y s ” a h a u s e e r i a t s t i a e u d u a n o m m o n s n t e o m t s i o l v i t n y u a e c n r e r - s e l t i u t i n l e c w y a f n q r c a s f n d e t l s d m t d i e , l t e o g r o n h d u o o r e e t r b o s e o ’ y o u r d ( r a r i e r t a y s r w m a A , r o h b m m d p m e o a d . D t i s w i i S e s k o c f d n i e a e b t h s u y p s c r a m l i t y e a c d n n c s ν e t o n e i e d e m k a u e f t e r a e p e e o i r e W g o l u t t o o e e r e s n l e . n n h a c l t l l e u r t c S p l m e f o c s H i t n i s i l a a n r f a l . m a h e o y d e w l o t e o e e c h n a l b c h e e h e t s t t H g o s m i t e i b d s s l i r a e a n t e i i l w w t y e a c o d C a e n o a b t h w u r p S t o s o o d u a c t u : e o O s v r o l t c y s n p g t s e a b e e r i u h e i i e r s e n a a p g r M c e l a s r r e , w n e [ a d h n e c , t F u y r g a d s d a r e o o . , o o e c 1 c e e t r r i r y r s r r l o h h d n g o u e e e c e e s s s c n b - e n ------f a , ] , e , . i o p y w b A B t r o s s e f l e i o p d r n s i i d o i w n t l t u u l e e e n o o h h o s m g a r m m i l g s o f t i e c a n m f t c q v o m d o m u d M m T a f c s t h a a H s h p s n l n t r f n i a o h e o h n e r h r f o w o h y i a u o c r f d n n e u y i o p i o a a o S M o d h a e d c o i d r o o o o e s a n a n m - o c i e n s o r L q d n d d t l c p e s e e r t o s e n 2 k m , t T t n i e r d l l t d d d d s r r d n d i u a l e m a fi u o n r s o p i r s y e u n e , g l o t e e i c m e t u n s e e e e d t c t n h o i e m g , l e o a n t t m S r r r e m e c r m 2 o n n t a n e t s d o v l l l l h p c l c o a s d t c r a o f , e Q t y A e o o e r , , d 0 f e . r t p t a e k s t w t e t p t e a a o r o o r u a t m s a n d ( r i v 0 a u t a e b b m S t i e r i o p n C n o s , o e i w r n l T r a e f w n c e o o i o t s o 8 r e h y s o s m o 5 h o d e e t g c l r p p i r y t d e h r d f a a d a a n ) D a n r r o q D d n h h s e n i p 0 s u t e u l f l u e r t s o t n a t l y k i c t i o e t i M o t u r t i 0 p i v e i r i p o r s s s n o r e e o - o o e d h t t s t n t s c l d r t n , c g r - e l o 5 l a e e a o a p r s f v p h f o e i n a e t e o i i u i h l v h o i o e l r l g i i h s a f o r t , o e m a i t n f r k w h h e e d e r t s m l I w r e l o ∗ l e t u e d u a k a e i t e m t s k t n l r y n h i . . s e e o v e c d h I e o n l a h c A n n m e h t e n s p p e t h t i t i u s p s . e s o n s a r t c w a t l m , d t l u t H i m e n d h f y t d t , l e c e A e u . h e c l m n o a t n i c t u a p i a b o h e a c p w t e i p l a o c n n n a a e s ) c h l e a l o e m a H t e y o f n I r h n h r r y t e o n v e m t r . n l n i a p T t r i t t u m r g n l s e e w i d l t a c r n s d h d t s p e r e e p r s a s d o e t t r s e p fi r t a e r h n h o o y i . o f r b H t l f a o s o e e i r r t p e o t d h h e u e e r a t l o o e n a s o t e r h b d . a o w i a m o l o i i e n n d a h p h h a r o s h s e s r r u o s n a c E n s e t o v g r r e a f s n t c T p n r t e u m u d e s a r e m u i l e f h n e i g m m r r i c w h i m t r a o n t e , c c s o o n n P m m c d i e o h o o l d d i S o e e s s e H t e e , d s h e l a l m o d o c n i b e e e s t e s u o u e i b y e m o h h a . n h e s , w u s l o p a e m s a x a t r i d n q c l E p n d y e s r u l u h e o v j e o e n t S e a l w l d ( r p l n a i i o e t w u e i s s a t a o b e v a y r p s p m s b e d e L s g a n o i B n r s i l e r c d o a n f r i e l m r ) t e u b h u e l e l c e m l e e j u e d t l u k s e g l c O t o , r e e e l s t 3 s y d e e p m e l r s / , i r a h w e t l . e i b x p g v i , n h t c d n . n c v , a s o e H r q g N s e o w a i h - a p e e [ e b b a l p t o h d a e t c e e c l b a o e n I n e 5 c i u r f s t n U m . a n h “ u o v e m r e e e m n h n h r o o a d p i o g s n t a e s ] o s e r t d d e r t w M h y r . e e r g h n i u a a m n m d i t . r t a e a s o p s p v o o a i s c y e i u w d o o l y e a e h o l n , h m l r f t r p s b e e d s h e o n . o f g p d [ d e t o n O n T n m p i g i e b h e i i 6 u m n v o b l s b r t l e h o t e n c o e s e r t e t “ d e d m e r o e l o ] F t u i l h e b t o i b s D h w t o m n h t o n l e o n n t e f o n a o h r o e g . n l c y h c o f t u e t a f o e d e s m e s g p a t l c n t e d a o r d y v p E i r r t o h l l s e r A t e o o n l n d a i c o h e a t e t t o h i o i r f e a l A p c e h i d a s p r n b n t L e l a a a h l o l a a l n c c D e o w t l o r a m a e l s y y , p l i s e o b y a l o y e g s k l k e r a e t h y s d t b l s , , , , p r s a r l i r P w u t i e . d n t e o g r e s l r u y n i i e e t g e h t j h h t l t o i g l s o r c u i e - a n h p h a e q - r m I c t y o l h c a 0 e ” o f o o i a d b g u t t t c . o s r i t l , i u r n a n 5 e t w m u p r n h l s e h a t o i o u a w o e S t a l m w i - a - n r o p a s e d d e e n b s g a s e 0 e n c i r M t e p l u , t a s v n a e f , 5 i m i i t s e e s p d i c r e o n s c r r s e o a g r d o i n s o H / s d ” b l a .

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. 2 model by twists through ±π in a ribbon. For convenience tremely simple, being defined by only a single property let us denote a twist through π as a “dum”, and a twist (i.e. whether they twist through +π or −π). This may through −π as a “dee” (U and E for short, after Tweedle- be interpreted as representing exactly two levels of sub- dum and Tweedle-dee [7]). Generically we refer to such structure within quarks and leptons. twists by the somewhat whimsical name “tweedles”[8]. The helon model also improves on the original rishon We hope to deduce the properties of quarks and leptons model by explaining why the ordering of helons (which and their interactions from the behaviour of their con- are analogous to rishons) should matter. There are three stituent tweedles, and to do so we shall employ a set of permutations of any triplet with a single “odd-man-out”. assumptions that govern their behaviour: Without braiding we cannot distinguish these permuta- 1) Unordered pairing: Tweedles combine in pairs, so tions, by the rotational invariance argument above. How- that their total twist is 0 modulo 2π, and the ordering ever, if we allow braiding, the strands (and hence per- of tweedles within a pair is unimportant. The three pos- mutations) become distinct, in general. We may asso- sible combinations of UU, EE, and UE ≡ EU can be ciate these permutations with the three colour charges of represented as ribbons bearing twists through the angles QCD, just as was done in the rishon model, and write the +2π, −2π, and 0 respectively. A twist through ±2π is helons in ordered triplets for convenience. The possible interpreted as an electric charge of ±e/3. We shall refer combinations and their equivalent quarks are as follows to such pairs of tweedles as helons (evoking their helical (subscripts denote colour): structure) and denote the three types of helons by H+, H+H+H0 (uB) H+H0H+ (uG) H0H+H+ (uR) H−, and H0. 2) Helons bind into triplets: Helons are bound into H0H0H+ (dB ) H0H+H0 (dG) H+H0H0 (dR) triplets by a mechanism which we represent as the tops H−H−H0 (uB) H−H0H− (uG) H0H−H− (uR) of each strand being connected to each other, and the bot- H0H0H− (dB ) H0H−H0 (dG) H−H0H0 (dR). toms of each strand being similarly connected. A triplet while the leptons are: of helons may split in half, in which case a new connection + − forms at the top or bottom of each resulting triplet. The H+H+H+ (e ) H0H0H0 (νe) H−H−H− (e ) reverse process may also occur when two triplets merge Note that in this scheme we have created neutrinos, but to form one triplet, in which case the connection at the not anti-neutrinos. This has occurred because the H0 top of one triplet and the bottom of the other triplet “an- helon is its own anti-particle. This apparent problem will nihilate” each other. be turned to our advantage in Section III. Note also that The arrangement of three helons joined at the top and we have not ascribed any of the usual quantum numbers bottom is equivalent to two parallel disks connected by (such as mass or spin [13]) to the helons. It is our expec- a triplet of strands. In the simplest case, such an ar- tation that spin, mass, hypercharge and other quantities rangement is invariant under rotations through angles emerge dynamically, just like electric charge, as we ar- of 2π/3, making it impossible to distinguish the strands range helons into more complex patterns. without arbitrarily labelling or colouring them. However It is interesting to note that three helons seems to be we can envisage the three strands crossing over or under the minimum number from which a stable, non-trivial each other to form a braid. The three strands can then structure can be formed. By stable, we mean that a be distinguished by their relative crossings. We will ar- physical representation of a braid on three strands (e.g. gue below that braided triplets represent fermions, while made from strips of fabric) cannot in general be smoothly unbraided triplets provide the simplest way to represent deformed into a simpler structure. By contrast, such a gauge bosons. physical model with only two strands can always be un- 3) No charge mixing: When constructing braided twisted. triplets, we will not allow H+ and H− helons in the same triplet. H+ and H0 mixing, and H− and H0 mixing are allowed. III. PHENOMENOLOGY OF THE HELON 4) Integer charge: All unbraided triplets must carry MODEL integer electric charge. Assumption 1) is unique to the helon model, as it reflects A braid on n strands is said to be an element of the the composite nature of helons, however assumptions 2), 3) and 4) are merely restatements in a different context of braid group Bn. Given any element of Bn, it is also pos- sible to create a corresponding anti-braid, which is the assumptions that Shupe and Harari made in their work. We will require one further assumption, however it will top-to-bottom mirror image of that braid, and is also an element of B (it is therefore merely a matter of conven- be left until later, as it can be better understood in the n proper context. tion what we call a braid or an anti-braid). If we join the strands of a braid with those of its top-to-bottom In terms of the number of fundamental objects, the helon model is more economical than even the rishon model, al- mirror image (i.e. we take the braid product [9]) we ob- tain a structure which can be deformed into the trivial beit at the cost of allowing helons to be composite. This seems a reasonable price to pay, as the tweedles are ex- braid. Conversely, if the strands of a trivial braid are crossed so as to form a braid, an anti-braid must also be 3 3

− FIG. 1: The fermions formed by adding zero, one, two or FIG. 2: A representation of the decay µ → νµ +e +νe, show- three charges to a neutral braid. Charged fermions come in ing how the substructure of fermions and bosons demands two handedness states each, while ν and ν come in only one that charged leptons decay to neutrinos of→the same g−enera- FIG. 1: The fermions efaocrhm. (e3d) debnyoteasdthdaitntgherzeearroe,thorneeep, otsswibole operrmutatFioInGs,. 2: tiAon.representation of the decay µ νµ +e +νe, show- three charges to a neutirdaenl tbifireadiads.thCe qhuaarrkgecodloufersr.mTihoenbsancdosmatetoipnand boitntogmhow the substructure of fermions and bosons demands two handedness states ereapcrhes,enwt hthielebiνndainngdofνhecloonms. e in only one that charged leptons decay to neutrinos of the same genera- each. (3) denotes that there are three possible permutations, tion. from the same sub-components (by contrast the rishon identified as the quark colours. The bands at top and bottom model used neutral rishons for the νe and neutral anti- represent the binding offohrmeleodn. sT. hus if we assign a quantity β = +1 to braids rishons for the νe). and β = −1 to anti-braids, NB = β is conserved in If we perform C and P operations on any braid (except splitting and joining operations (wh!ere the sum is taken a ν or ν, on which we cannot perform P) we obtain the over all braids and anti-braids present). These profproerm- thbreaisdadmiageonsaullby-ocpopmosipteoint einnFtisgu(rbe y1. cWoenmtraaysdtefitnhe ea rishon ties are reminiscent of the relationship between partmicloesdel fuusrtehder nopeeurattriaonl wrihsichhocnonssifsotsrofthroetatνinegaanbdraidnecluoctkr-al anti- formed. Thus if we aasnsdiganntai-pqarutiaclnest,itayndβso=it i+s n1attuoralbtroauisdesthe torpi-stoh-onswfioserotrhaentiν-cel)o.ckwise through π, and reversing the sign − bottom mirroring of braids as a model for particle–anti- of all charges. This operation will be called T, and we and β = 1 to anti-bpraartiidclse,inNteBrch=ange, orβCisinvceorsniosne.rvInedadidnition, gIifvewn e pneortfeotrhmat pCerfaornmdingPCo, Pp,earnadtTioinnsanoynoradneryonbarabirdaid(except splitting and joining oapnyerealetmioents o(fwBhn!e, rietstlhefet-tsou-rmightismtiarrkoer nimage amaνy or lνea,voesnthwathbircahidwunechcaanngendo. t perform P) we obtain the over all braids and anbteif-obrrmaeidd. sWperweisllencatl)l .theTsehtehseelepft-rhoapndeerd-and rbigrhat-id dHiaagvionng acolnlystroupctpedostihteeqiutarikns Fanidgulerpeton1s., wWe enomw atuyrndefine a ties are reminiscent ofhatnhdeedrfeolramtsioofnasbhriapidb. Iettsweeemesnnaptauratlictoleesquate tfhuersteheroourpeatrtaetnitoionn wtohtihcehirciontnesraisctisonosf vrioattahteinelgecatrobwreaaikd clock- with particles and anti-particles having positive and neg- and colour forces. and anti-particles, andatisvoe hiteliicsityn.atFurormalthteo duissceustshioen taobopv-et,oa-ll fermwioinsse orTahnetEi-lceclotrcokwweaiskeInthterroauctgiohnπW, easnhdallrbeevgienrsbiyncgont-he sign bottom mirroring of barnadidthseiar sanati-mpaortdicelels faorer rpeparretsiecnlteed–abny tbir-aids wohfichall cshtraurcgtiensg.thTe hbiossonospoefrathteioenlecwtroilwleabkeinctaerlalectdionT, ,γ,and we + − 0 + − particle interchange, oarre CeleminenvtesrosfiBon3.. In addition, given note thWat ,pWerfo, ramndinZg. CT,heP,WanadndTWin amnayy boerdreegrarodneda braid Let us construct the first-generation fermions with pos- as a triplet of H+s and a triplet of H−s respectively. We any element of Bn, iittisvelechfta-rgteo-arsigshhotwnmoinrrtohre riimghtagofeFmiguarye 1 (weleaarve es tchanatcrebarteaindeuutrnalcbhoasonngsefdro.m a triplet of similar helons be formed. We will causllingthaebsaesicthberailde,flti-khe athnadteudsedantodplraiigt hhta-ir, for Hilluasv- ingincotwnoswtrauysc.tOednetihs easqautarirpklest aofnudntlweipstteodnhse,lownse, tnheow turn handed forms of a braitdra.tiIvte speuerpmossesn, ahotwuervaelr taon earqbuitaratrey tnhuemsbeer of corousrs- attoethnetrioans atotriptlheteiorf “icnotuenrtaerc-twioisntesd”vhiaelotnhs e(thealtecist,roweak ings is possible). This yields the positron, up quark, each helon carries explicit left-handed and right-handed with particles and antai-nptia-drotwicnleqsuahrka,vainndg apnotis-inteiuvterinaon. dNnoewg-let us acnond- coltowuisrts)f.orWceess.hall claim that the former is the photon, ative helicity. From stthruectdtihsecunsesgiaotnivelayb-cohavreg,edalfelrmfeiornms iobynstakingTthhee Etlheecltarttoerwisetahke ZI0n(tweermaacytailosonspWecuelasthe athllatbdeegfoirnm-by con- and their anti-particletospa-troe-bortetpomresmeinrrtoerdimbaygesborfaitdhes pwoshitiicvhely chasrtgreud ctiningg atnheuntbwoisstoednsheloofn tinhtoe aecleoucnttreorw-tweiastkedinhetleorna, cortion, γ, fermions. We have constructed the positively charge+d vi−ce-versa, acc0ounts for the W+einberg mix−ing between the are elements of B3. fermions by adding positive charges (dees) to a rWight- , WZ0 a, nadntdhe Zpho.toTn h[1e4])W. WhaatnsdetsWbosonms aapyarbt efrormegarded Let us construct the fiharnsdte-dgebnraeidr,aatniodnthefierramntiio-pnarstiwcleisthby padodsi-ng negatsivae trifperlmetionosf (Hi.e+. ssoanthdataatγripisledtistoinfcHt f−rosmraesnpeeutcrtinivo,ely. We ± ± itive charge as shownchoanrgetsh(edumrisg)htto oa fleFft-ihgaunrdeed1an(twi-beraaidr.eThis cdaecni- creaantdeanWeutirsadlisbtioncstofnrosmfraonme a) istrthipatlettheosftrsainmd iplearr- helons sion was of course completely arbitrary. We can also add mutation induced by the braid that forms a boson is the using a basic braid, likdeesthtoaat luefste-hdantdoedpblraaiitd hanadird,ufmosrtoillaursi-ght-hainndetdwo iwdeanytisty. pOernmeutiastioans. aThtersiipmlpeltesot bfruaindttwhaitsftueldfillshtehlios ns, the trative purposes, howaenvtei-rbraaind.aIfrwbeitdroatrhyis,nwuemcrebaeteraollfthcerpoossss-ible chaortghe-er acsriteariotnriips ltehte torifvia“lcboruaind,taesr-intwFiigs.te3.d” helons (that is, ings is possible). Thciasrryyinieglbdrsaidtshinetwpoodsiiffterroennt,haunpdedqnuesasrskta,tes exaecatclyh heAllol nintceraarcrtioenss ebxetpwleiecnithelleofnts-chaannbdeevdiewaenddasrciugthtitn-ghanded once, but following the same procedure for the uncharged or joining operations, in which twists (tweedles) may be anti-down quark, andbraaidnst(ii-.ne.enueturtrininoos. andNaonwti-nleeuttriunoss)cwoonul-d meantwdui-sts).excWhanegesdhbaeltlwceelnaihmelonths.aTthtehseeopfoerramtioenrs disefinthe ebap- hoton, 0 struct the negativelyp-lcichaatirnggetdhemf,esrimnceiothniss sebcyondtpaakirinofgneutthreal leptotnhseis lattseicrviesrttichese fZor he(lwoneinmtearayctaiolnsso. sBpyecoumlabitneintghtahtreedeform- top-to-bottom mirroridiemntaicgaletso othfe tfihrset ppaior,sirtoitvateeldythcrhouagrhge±dπ. In oitnhger an ouf nthtewseisbtaesdic hheelolnonveritnicteos ian pcaoraullneltewre-ctownisstrtuecdt thheelon, or words, to avoid double-counting we can only construct basic vertices of the electroweak interaction (Figure 4). fermions. We have ctohne s(tarnuti-c)tneedutritnhoeinpaos(riitgihvte-)llyeft-chhaandregdedform, wvhicile-veCrsraos,siancgcsoyumnmtestrfioesrotfhteheWheeloinnbverrtgicems aixutionmgabticeatlwlyeen the 0 fermions by adding paollstihteivotehecrhfearmrgioenss c(odmeeeisn)bottoh leaft-raingdhrti-ght-haZndedandimtphlye tphehuostuoanl cr[o1s4si]n)g. syWmmheattriesseftosr tbhoe seolenctsroawpeaakrt from handed braid, and thefiorrmasn.tTi-hpisaprlteiacsliengs rbeysulatdisdaindigrenctecgoantseiqvueence offetrhme ionvsert(icie.es.. so that a γ is distinct from a neutrino, charges (dums) to a flaecftt-thhatndwe dconasntrtui-cbt rtaheidn.eutTrihnoisanddecain-ti-neuatrnindo a WWe±cains dreipsrteisnenctthfirgohmer gaennerea±tio)nisfertmhiaotnsthbye asltlorwa-nd per- sion was of course completely arbitrary. We can also add mutation induced by the braid that forms a boson is the dees to a left-handed braid and dums to a right-handed identity permutation. The simplest braid that fulfills this anti-braid. If we do this, we create all the possible charge- criterion is the trivial braid, as in Fig. 3. carrying braids in two different handedness states exactly All interactions between helons can be viewed as cutting once, but following the same procedure for the uncharged or joining operations, in which twists (tweedles) may be braids (i.e. neutrinos and anti-neutrinos) would mean du- exchanged between helons. These operations define ba- plicating them, since this second pair of neutral leptons is sic vertices for helon interactions. By combining three identical to the first pair, rotated through ±π. In other of these basic helon vertices in parallel we construct the words, to avoid double-counting we can only construct basic vertices of the electroweak interaction (Figure 4). the (anti-)neutrino in a (right-)left-handed form, while Crossing symmetries of the helon vertices automatically all the other fermions come in both left- and right-handed imply the usual crossing symmetries for the electroweak forms. This pleasing result is a direct consequence of the vertices. fact that we construct the neutrino and anti-neutrino We can represent higher generation fermions by allow- 3

− FIG. 1: The fermions formed by adding zero, Avrinone, tw o or FIG. 2: A represenBilson-Thompsontation of the decay µ → νµ +e +νe, show- three charges to a neutral braid. Charged fermions come in ing how the substructure of fermions and bosons demands two handedness states each, while ν and ν come in only one that charged leptons decay to neutrinos of the same genera- each. (3) denotes that there are three possible permComparingutations, ti oTopologicaln. Formalisms identified as the quark colours. The bands at top and bottom represent the binding of helons. from the same sub-components (by contrast the rishon model used neutral rishons for the νe and neutral anti- formed. Thus if we assign a quantity β = +1 to braids rishons for the νe). and β = −1 to anti-braids, NB = β is conserved in If we perform C and P operations on any braid (except splitting and joining operations (wh!ere the sum is taken a ν or ν, on which we cannot perform P) we obtain the over all braids and anti-braids present). These proper- braid diagonally opposite it in Figure 1. We may define a ties are reminiscent of the relationship between particles further operation which consists of rotating a braid clock- and anti-particles, and so it is natural to use the top-to- wise or anti-clockwise through π, and reversing the sign bottom mirroring of braids as a model for particle–anti- of all charges. This operation will be called T, and we particle interchange, or C inversion. In addition, given note that performing C, P, and T in any order on a braid any element of Bn, its left-to-right mirror image may leaves that braid unchanged. be formed. We will call these the left-handed and right- Having constructed the quarks and leptons, we now turn handed forms of a braid. It seems natural to equate these our attention to their interactions via the electroweak with particles and anti-particles having positive and neg- and colour forces. ative helicity. From the discussion above, all fermions The Electroweak Interaction We shall begin by con- and their anti-particles are represented by braids which structing the bosons of the electroweak interaction, γ, + − 0 + − are elements of B3. W , W , and Z . The W and W may be regarded Let us construct the first-generation fermions with pos- as a triplet of H+s and a triplet of H−s respectively. We itive charge as shown on the right of Figure 1 (we are can create neutral bosons from a triplet of similar helons using a basic braid, like that used to plait hair, for illus- in two ways. One is as a triplet of untwisted helons, the trative purposes, however an arbitrary number of cross- other as a triplet of “counter-twisted” helons (that is, ings is possible). This yields the positron, up quark, each helon carries explicit left-handed and right-handed anti-down quark, and anti-neutrino. Now let us con- twists). We shall claim that the former is the photon, struct the negatively-charged fermions by taking the the latter is the Z0 (we may also speculate that deform- top-to-bottom mirror images of the positively charged ing an untwisted helon into a counter-twisted helon, or fermions. We have constructed the positively charged vice-versa, accounts for the Weinberg mixing between the fermions by adding positive charges (dees) to a right- Z0 and the photon [14]). What sets bosons apart from handed braid, and their anti-particles by adding negative fermions (i.e. so that a γ is distinct from a neutrino, charges (dums) to a left-handed anti-braid. This deci- and a W ± is distinct from an e±) is that the strand per- sion was of course completely arbitrary. We can also add mutation induced by the braid that forms a boson is the dees to a left-handed braid and dums to a right-handed identity permutation. The simplest braid that fulfills this anti-braid. If we do this, we create all the possible charge- criterion is the trivial braid, as in Fig. 3. carrying braids in two different handedness states exactly All interactions between helons can be viewed as cutting once, but following the same procedure for the uncharged or joining operations, in which twists (tweedles) may be braids (i.e. neutrinos and anti-neutrinos) would mean du- exchanged between helons. These operations define ba- plicating them, since this second pair of neutral leptons is sic vertices for helon interactions. By combining three identical to the first pair, rotated through ±π. In other of these basic helon vertices in parallel we construct the words, to avoid double-counting we can only construct basic vertices of the electroweak interaction (Figure 4). the (anti-)neutrino in a (right-)left-handed form, while Crossing symmetries of the helon vertices automatically all the other fermions come in both left- and right-handed imply the usual crossing symmetries for the electroweak forms. This pleasing result is a direct consequence of the vertices. fact that we construct the neutrino and anti-neutrino We can represent higher generation fermions by allow- 3 3

− FIG. 1: The fermions formed by adding zero, one, two or FIG. 2: A representation of the decay µ → νµ +e +νe, show- − FItGhr.ee1:chaTrhgees fteorma inoenustrfaolrmbreadid.byChaadrdgeindgfezrmerioo,nsoncoem, etwino or ing FhoIwG.t2h:eAsurbesptruescetunrteatoiof nfeorfmtihonesdeacnadybµos→onsνµd+emean+dsνe, show- thtrweoe hcahnadregdensetsos satanteesuteraaclh,bwrahiidle. νCahnadrgνedcofmereminioonnslycoomnee in thatinchgarhgoewd ltehpetonsusbdsetcrauycttuorneeuotfrifneorsmoiof nthseasnadmebogseonnersa-demands tweoach.an(3d)eddneneostsesttahtaets tehaecrhe,awrehtihlereνe paonsdsibνlecopmeremuintaotinolnys,one tion.that charged leptons decay to neutrinos of the same genera- eaidchen. t(ifi3e)ddaesntohteesqutharakt ctohloeruersa. rTehtehbreaendpsoasstitbolpe apnedrmboutttaotmions, tion. idreenptriefiseedntatshtehbeinqduianrgkocfohleoluornss.. The bands at top and bottom represent the binding of helons. from the same sub-components (by contrast the rishon model used neutral rishons for the ν and neutral anti- from the same sub-componentes (by contrast the rishon formed. Thus if we assign a quantity β = +1 to braids rishons for the νe). model used neutral rishons for the νe and neutral anti- and β = −1 to anti-braids, NB = β is conserved in If we perform C and P operations on any braid (except fosrpmlitetdin. gTahnudsjoifinwineg aospseirgantioanqsu(awnhteirteytβhe=su+m1istotakbernaids a ν roirshνo,nosnfowrhitchhewνee)c.annot perform P) we obtain the − ! anodverβa=ll bra1idtsoaanndtia-nbtria-bidras,idNs Bpre=sent).βTihsecseonpsreorpveerd- in braiIdf dwiaegponearflolyrmoppCosaitneditPinoFpigeruareti1o.nsWoenmaanyydebfirnaeida (except sptliietstianrge raenmdinjiosicneinntgofoptheerarteiloantison(swhhip!erbeettwheesnupmaritsictleasken furtaheνr ooprerνa,tioonn wwhhiicchhcwonesicsatsnonfortotpaetrinfogrambrPa)idwcleocokb-tain the ovaenrdaalnltbi-rpaairdtisclaens,danadntsio-birtaiisdnsapturreaslentot)u.seTthheesteopp-rtoop- er- wiseborariadndtii-acgloocnkawlliyseotphproousgithe πit, ianndFirgeuvreers1in.gWtheemsiagyndefine a tibesotatoremrmemirirnoirsicnegnotfobfrtahidesraeslaatimonosdhelipfobreptawreteicnlep–aanrtii-cles of aflul rcthhaerrgeosp. eTrahtiisonopwehraictihoncownislilsbtseocfarlloetdatTin, ganadbrwaeid clock- anpdaratinctlei-pinatretricchlaesn,gea,nodrsCo iitnvisernsiaotnu.raInl taodudisteiotnh,egtivoepn-to- notewtihsaetoprerafnotrim-cinlogcCkw, Pis,eanthdrTouignhanπy, oarndderroenvearsbirnagidthe sign boatntyomelemmeirnrtoroifngBno,f bitrsaildefst-atos-raigmhtodmeilrrfoorr ipmaartgieclme–aaynti- leavoefs tahllatchbararigdesu.ncThahnisgeodp. eration will be called T, and we pabretifcolremiendt.erWcheawniglle,caolrl tChesinevtehresiloenft.-haInndaedddaintdionri,ghgti-ven HavninogtecotnhsattrupcetrefdortmheinqguaCrk,sPa,nadnldepTtoinns,awnye noorwdetruornn a braid handed forms of a braid. It seems natural to equate these our attention to their interactions via the electroweak any element of Bn, its left-to-right mirror image may leaves that braid unchanged. bewiftohrmpaerdti.clWes eanwdilalnctia-lplatrhtiecslesthhaevilnegftp-hoasintdiveedaannddnergig-ht- andHcoalvoiunrgfocorcnesst.ructed the quarks and leptons, we now turn ative helicity. From the discussion above, all fermions The Electroweak Interaction We shall begin by con- handed and thfeoirrmanstoi-fparbtircaleids .arIet srepemresennatetdurbayl tboraeiqdus awtheitchese struocutirngatttheentbioosnontsootfhtehier einletcetrraocwteioaknsinvtiearacthtieone,leγc,troweak with particles and anti-particles having positive and neg- +and −colour fo0rces. + − are elements of B3. W , W , and Z . The W and W may be regarded atLivete uhsecliocnitsytr.ucFtrtohme fitrhset-gdeinsceruastsiioonnfearbmoivoen,s awlilthfeprmosi-ons as aTtrhipeleEt loefcHtr+oswaenadka ItnriptelertaocftHio−ns Wresepeschtaivllelbye. gWine by con- anitdivethcehiraragnetai-spsahrotwicnlesonartehererpigrhesteonfteFdigbuyreb1ra(iwdes awrheich cansctrreuactteineguttrhael bboossoonnssfromf tahetrieplleecttorfowsimeaiklarinhteelorancstion, γ, + − 0 + − aruesienlgema ebnatssicobfrBai3d., like that used to plait hair, for illus- in twWo w,aWys. O, naendis Zas a. tTrihpeletWof uanntwdisWted hmeloanys,bteheregarded Letrtatuisvecopnursptrousecst, thhoewefivresrt-agnenaerrbaittriaorny fneurmbioenrsofwcirtohssp-os- othears aas tariptrleiptleotf oHf +“scoaunndtear-twriipslteetd”ofhHelo−nssr(etshpaetctiisv, ely. We itinvgescihsaprgoessiabsles).howTnhisonyietlhdes rtihgehtpoosfitFroing,uruep 1qu(warek,are eachcahnelcorneacaterrnieesuetxrpallicbitosleofnt-shfarnodmedaatnrdiprleigthot-fhsainmdieldar helons usainntgi-daobwansiqcubarraki,da,nlidkeanthtia-nteuusteridnot.o pNlaowit hleatiru,sfocronil-lus- twisitns)t.wWo ewashyas.ll Oclnaiemisthaastathtreipfolertmoefr uisnttwheisptehdothonel,ons, the 0 trsattriuvcet ptuhreponseegsa,tihvoelwye-cvhearrgaendafrebrmitrioanrys nbuymtbakeirnogf tchreoss- the olatthteerr iassthae tZrip(lwete mofay“caolsuonstpere-ctuwlaitsetetdh”athdeelfoonrsm-(that is, intgosp-itso-bpootstsoibmlem).irrTorhiismaygiesldosf tthhee ppoosittirvoenly, cuhparqguedark, ing eaanchunhtweliosntedcahrerlioens einxtpoliacitcoluenftt-ehr-atnwdisetdedanhedlornig, hotr-handed anfetrim-dioownsn. qWuaerhka, vaencdonasntrtui-cnteeduttrhineop. osNitiovwelylecthaursgecdon- vicet-wveirsstas,).accWouentsshfaolrl tchleaiWmeitnhbaetrgtmheixifnogrmbeetrweisenththeephoton, fermions by adding positive charges (dees) to a right- Z0 and the photon [14]). What sets bosons apart from struct the negatively-charged fermions by taking the the latter is the Z0 (we may also speculate that deform- handed braid, and their anti-particles by adding negative fermions (i.e. so that a γ is distinct from a neutrino, tocph-atrog-ebso(tdtuoms)mtiorraorleifmt-haagnedsedofatnhtie-bpraoisdi.tivTehlyis cdheacir-ged andinagWa±n ius ndtiswtinstcetdfrohmeloann ien±t)oisathcoatunthters-ttrwainsdtepderh-elon, or fermions. We have constructed the positively charged vice-versa, accounts for the Weinberg mixing between the sion was of course completely arbitrary. We can also add mutat0ion induced by the braid that forms a boson is the fedremesiotnosablyefta-hdadnindgedpborsaiitdivaendchdaurmgessto(daeerisg)htt-ohaandreidght- idenZtityapnedrmtuhteatpiohno.toTnhe[1s4im])p. leWst hbaratidsetthsatbfouslofinlls tahpisart from haantdie-bdrabirda.idIf,waendotthheiisr, wanetcir-epaatretaiclletshebypoasdsidbilnegchnaerggae-tive criteferiromniiosntshe(it.rei.viaslobrtahiadt, aas iγn iFsigd.i3s.tinct from a neutrino, ± ± chcaarrgyeisng(dburamidss)intotwao ldeifffte-hreanntdheadndaendtnie-bssrastidat.esTexhaisctldyeci- All ainntderacWtionsibsedtwiseteinnchteflroonms cann bee v)iieswtehdaatstchuettsitnrgand per- sioonncew,absuotffoclolouwrsinegctohme psalemtelpyroacrebditurraerfyo.r Wthe ucnacnhaalrsgoedadd or jominuitnagtioopneriantdiouncse,dinbywhtihceh btwraisidts t(htwateefdolrems)smaabyobseon is the debersaitdos (ai.el.efnte-huatrnindoesdabndraaindtia-nedutdruinmoss)twooualdrimgheta-nhdaun-ded exchidanengetditybeptewremenuthaetlioonns.. TThheesseimoppelerasttibonrasidetfihnaetbfua-lfills this anptlic-batrianigd.thIefmw,esidnoceththisi,s wseecocnredaptaeiralolftnheeuptroaslsliebplteocnhsaisrge- sic vcerritteicreios nfoirs htehleontriinvtiearlabctriaoinds,. aBs yincFomigb.in3i.ng three ± caidrreynitnicgalbtroaitdhseinfirtswt opadiirff, erroetnatehdatnhdreodugnhess πst.aItnesoetxhaerctly of thAelsleinbtaesricachteiloonnsvbeerttiwceeseninhpealoranlslecl awne bcoenvsiterwucetdthaes cutting onwcoer,dbs,uttofoalvloowidindgouthbeles-caomunetpinrgocwede ucraenfoornltyhecounnsctrhuacrtged basiocrvjeoritnicinesgoofptehreateiloecntsr,owineawkhiincthertawctisiotns ((tFwigeuerdele4s)). may be brtahieds(a(ni.tei-.)nneeuuttrrininoosinanad(aringthit--n)eleuftt-rhinanods)edwofourlmd,mwehainledu- Croessxicnhgasnygmemd ebtreitews eoefnthheelhoenlosn. vTehrteisceesoapuetroamtiaotnicsaldlyefine ba- all the other fermions come in both left- and right-handed imply the usual crossing symmetries for the electroweak plicating them, since this second pair of neutral leptons is sic vertices for helon interactions. By combining three forms. This pleasing result is a direct consequ±ence of the vertices. idfeanctictahlattowtehecofinrsstrtupctairth, eronteautterdintohraonudghanti-πn.euItnrinoother We ocfanthreesperebsaesnitc hhieglhoenr vgenrteircaetsioinn fperamraiollnesl wbye acollnowst-ruct the words, to avoid double-counting we can only construct basic vertices of the electroweak interaction (Figure 4). the (anti-)neutrino in a (right-)left-handed form, while Crossing symmetries of the helon vertices automatically all the other fermions come in both left- and right-handed imply the usual crossing symmetries for the electroweak forms. This pleasing result is a direct consequence of the vertices. fact that we construct the neutrino and anti-neutrino We can represent higher generation fermions by allow- 4 4

FIG. 3: The bosons of the electroweak interaction. Notice FIG. 3:0 The bosons of the electroweak interaction. Notice thatthtahtetZhe Za0ndantdhtehpe hpohtootonnccaann ddeeforrmm iinnttooeaecahchotohtehr.er. FIGF. I4G: . T4h:e Tchhaergcehtarragnesfetrrrainngsfevrerritnexg (vTeVrt)exan(dTtVhe) paon-d the po- larislianrgisvinergtevxer(tPeVx)(aPreVb)aasirce hbealosnicvheretliocnes vfreormticweshifcrhomthewhich the ing itnhgethtehrtehereheehloelnosnstotoccrroossss iinn moorree ccoommpplicliactaetdedpapt-at- elecetrleocwteraokwbeaaskicbvaesritcicevserctaincebsecfaonrmbeed.formed. terntse.rnTs.hTushuasllalflerfemrmioinonssmmaayy bbee viieewweeddaassaababsaicsincenu-eu- trintorin“ofra“mfraemweowrokr”k”totowwhhicichh eelleecctrriicc cchhaargrgesesaraereadadded,ed, and the structure of this framework determines to which andanwdillwthilelretfhoerreenfoortepanrottakpeaorftathke cooflotuhreinctoelroaucrtioinnt.eraction. and the structure of this framework determines to which The reader may verify that quarks will naturally form generation a given fermion belongs. If, say, a muon emits The reader may verify that quarks will naturally form genechraartgioenina tghievefonrmferomf iaonWb−elboonsgosn., Itfh,esaemy,isasimonuoonf tehmisits groupings with integer electric charge, which are colour- charge in the form of a W − boson, the emission of this neugtrraolu, pfoirngexsawmipthle itnhtreegeeqruealrekcstfroircmcihnagragep,rowthonic:h are colour- (trivially braided) boson does not affect the fermion’s neutral, for example three quarks forming a proton: (trivbiraalildyingbrsatriducetdu)reb. oTshoenredfooresthneomt uaoffnelcotsetshcehafregrembiuotn’s braiddoiensgnsottruchctaunrgee.gTenheerraetifoonr,e atnhde mmuusotntrlaonssefsorcmhairngtoe abut 0 0 H0 doems unoont-ncehuatnrigneo.gLenikeerwaitsieo,nt,aua-nndeutmriunsots mtraaynsofnolrymtrainntso- a dB + uR + uG = ⎡ H0 0 0 0 0⎤ H0 d + u + u =0 HH− 0 0 0 0 muofonr-mneiunttrointaou. pLaikrteiwcleissea,ntdaue-lenceturotnri-nnoeustmrinaoys omnalyy otnrlayns- B R G⎣ ⎡ ⎦ ⎤ formtrainstfoortmauinptoaretlieccltersonasn. dAesleimctilraorna-nrgeuumtreinntoaspmplaieys toonly 0 H− 0 0 0 H⎣0 0 0 ⎦H+ tranqsufaorrkms, ianlthooeulgehcttrhoenms.ecAhasniimsmilabry awrhgiuchmethnet Capabpilbiebso to angles become non-zero is not immediately apparent. No- + ⎡ H+ 0 0 ⎤ + ⎡ H0 0 0 ⎤ . (2) quarks, although th±e mechanism by which the Cabibbo 0 0 H0 0 0 H+ tice that since W bosons carry three like charges, such 0 H+ 0 0 H+ 0 angles become non-zero is not immediately apparent. No- ⎣+ ⎡ H+ 0⎦ 0⎣ ⎤ + ⎡ H0 ⎦0 0 ⎤ . (2) processes will ch±ange quark flavour but not colour. tice that since W bosons carry three like charges, such 0 H+ 0 0 H+ 0 The Colour Interaction We have thus far described In QCD there are e⎣ight gluons. Two⎦of t⎣hese are super- ⎦ proceleescstersowweialkl cihntaenragcetiqounas raks flspavliottuinrgbauntdnjootincionglouopr.era- positions of like colour/anti-colour states, which we may ThetioCnsoolonubrraIidnst.eWraectmioaynsWimeilahrlayverepthreussenftarcodloeusrcrinib-ed visuIanlisQeCaDs ttrhipelreetsawreitehigshotmgeluuontnwsi.stTedwostroafntdhse, saenadre super- teractions physically, this time as the formation of a sompeocsoituinotnesr-towf ilsitkeed csotrlaonudrs/a(snttria-ncdoslowuirthsteaxtpelsic,iwt phoicsh- we may electroweak interactions as splitting and joining opera- 0 tion“spaonncabkreasidtasc.k”Woef bmraaiydss. imEailcahrlsyetreopf rsetrsaenndtsctohlaotulriein- itivveiasnudalniseegaatisvettrwipislet,tsaswpeitrhthseodmesecruipnttiwonisotfetdhestZra,nds, and above). The remaining six are “pure” unlike colour/anti- teraocnteio-anbsovpeh-tyhsei-coathlleyr, cathnibse triemgaerdaesd atshae“sfourpmera-sttiroanndo”f, a some counter-twisted strands (strands with explicit pos- colour pairings, which we can visualise as the six permu- 0 “pawncitahkea stotataclkc”haorfgbe reaqiudasl. toEathceh ssuemt ooff tshtreacnhdasrgtehsaotnlie itive and negative twist, as per the description of the Z , tations of one H+, one H0, and one H−. each of its component strands. If we represent braids as above). The remaining six are “pure” unlike colour/anti- one-paebrmovuet-atthioen-omthaterricceasn, wbiethreagcahrdneodn-azesrao c“osmuppeorn-esnttrabne-d”, Hypercharge Recall that we assigned β = +1 to the colour pairings, which we can visualise as the six permu- withingaathoetlaoln,chwaercgaen eeqasuialyl rteoprtehseentsuthme cooflotuhreinctheraarcgteiosnon braids on the top row of Figure 1, and β = −1 to the eachbeotwf eitesn cfeormmpioonnseanstthstersaunmdso.f Ithf ewceorrreepspreosnednintgbmraaitdrsi- as bratidastioonnsthoefboontetoHm+ro,wo.neThHis0,effaencdtivoenlye dHis−ti.nguishes permceus.taFtoironinsmtaantcreicaesr,edwuitphqeuaacrhk,naonnd-zaenraonctoi-mredpoannetin-utpbe- betwHeyenpefrercmhiaorngsewiRthecaalnl etthpaotsiwtieveaasnsdignneetd nβega=tiv+e 1 to the ing caohmeblionnin, gwteocfaornmeaasnileyutrreapl rpeisoennctotuhlde bceolroeupreisnetnetreadcatsion chabrgrea.idTsoodnistihnegutioshp breotwweoenf Fquigaurkrse a1n,dalnepdtoβns=th−e 1 to the between fermions as the sum of the corresponding matri- numbbrearidosf oonddt-hmeenb-ootuttomwitrhoiwn .anTyhitsripeffleet,ctdiviveildyeddibsytinguishes ces. For inst0ance0a rHed0 up quark0, an0d anH0anti-red anti-up thebneutmwbeern offerhmeloionnsswwitihtihn aa tnrieptlept ossuigtgiveestsanitdselnf eats negative H+ 0 0 + H− 0 0 a useful quantity (taking a value of 0 for leptons and combinin⎡g to form a ne⎤utral ⎡pion could be ⎤represented as charge. To distinguish between quarks and leptons the − 1/3 for quarks). However this quantity does not distin- 0 H+ 0 0 H 0 number of odd-men-out within any triplet, divided by ⎣ ⎦ ⎣ ⎦ guish between a particle and its anti-particle. To rectify 0 0 H0 0 0 H0 the number of helons within a triplet suggests itself as 0 0 H0 + H0 this shortcoming we will define a new quantity, given by − ⎡ H+ =0 ⎡ 0H+⎤+ +H−⎡ H 0 0 0 0⎤ ⎤ (1) onea-thuirsdefuthleqnuuamnbtietryof(t“amkoinreg paosvitaivlue”e hoeflo0nsf,omr ilneupstons and 0 H+ 0 0 H+0 + HH−− 0 0 one1-t/h3irdfotrhqe unaurmkbse).r oHf o“wleessveprostihtiivse”quhaelnotnist.yWdeoesshanllot distin- ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ dengouteisthhibseqtwuaenetnitya bpyartthieclesyamnbdolitΩs.anTtoi-pclaarrtifiyc,leH. +To rectify where the sum of tw0o helons on0the righHt-0h+anHd-s0ide de- helotnhsisasrheocrotncsoidmeirnedg w“meowreillpodseifitinve”athnaenwHq0uahnetloitnys,, given by notes a super-strand composed of a pair of helons. In this which are “more positive” than H− helons. If N(H+) is = ⎡ H+ + H− 0 0 ⎤ (1) one-third the number of “more positive” helons, minus case all three super-strands have zero net charge. In this the number of H+ helons, N(H0) the number of H0s and 0 H+ + H− 0 one-third the number of “less positive” helons. We shall way hadrons can be regarded as a kind of superposition N(H−) the number of H−s within a triplet, and remem- ⎣ ⎦ denote this quantity by the symbol Ω. To clarify, H+ of quarks. We can now introduce our last assumption. bering that H+ and H− helons never occur within the where the sum of two helons on the right-hand-side de- 5) Charge equality:When two or more braids are samheeblornaisdeadretricpolents,idweermeday“wmroitree positive” than H0 helons, notesstaackseudp, etrh-esytrmanudstcocommpboinseedtoofparopdauicreotfhheesloamnse. tIontatlhis which are “more positive” than H− helons. If N(H+) is casechaalrlgtehorneeasllutpherree-ssturapenrd-sstrhaanvdes.zero net charge. In this the number1 of H+ hel1ons, N(H01) the number of H0s and Ω = β N(H+) + N(H−) − N(H0) . (3) wayIthaisdorobvnisoucsanthbate lreepgtaorndsetdrivaisalalykfiunldfillofthsius pcerritpeorisointi,on N(H−) t%he3 number o3f H−s with3in a tri&plet, and remem- of quarks. We can now introduce our last assumption. bering that H+ and H− helons never occur within the 5) Charge equality:When two or more braids are same braided triplet, we may write stacked, they must combine to produce the same total charge on all three super-strands. 1 1 1 Ω = β N(H+) + N(H−) − N(H0) . (3) It is obvious that leptons trivially fulfill this criterion, % 3 3 3 & Arxiv: June 2015

2 model by twists through ±π in a ribbon. For convenience tremely simple, being defined by only a single property let us denote a twist through π as a “dum”, and a twist (i.e. whether they twist through +π or −π). This may through −π as a “dee” (U and E for short, after Tweedle- be interpreted as representing exactly two levels of sub- dum and Tweedle-dee [7]). Generically we refer to such structure within quarks and leptons. twists by the somewhat whimsical name “tweedles”[8]. The helon model also improves on the original rishon We hope to deduce the properties of quarks and leptons model by explaining why the ordering of helons (which and their interactions from the behaviour of their con- are analogous to rishons) should matter. There are three stituent tweedles, and to do so we shall employ a set of permutations of any triplet with a single “odd-man-out”. assumptions that govern their behaviour: Without braiding we cannot distinguish these permuta- 1) Unordered pairing: Tweedles combine in pairs, so tions, by the rotational invariance argument above. How- that their total twist is 0 modulo 2π, and the ordering ever, if we allow braiding, the strands (and hence per- of tweedles within a pair is unimportant. The three pos- mutations) become distinct, in general. We may asso- sible combinations of UU, EE, and UE ≡ EU can be ciate these permutations with the three colour charges of represented as ribbons bearing twists through the angles QCD, just as was done in the rishon model, and write the +2π, −2π, and 0 respectively. A twist through ±2π is helons in ordered triplets for convenience. The possible interpreted as an electric charge of ±e/3. We shall refer combinations and their equivalent quarks are as follows to such pairs of tweedles as helons (evoking their helical (subscripts denote colour): structure) and denote the three types of helons by H+, H+H+H0 (uB) H+H0H+ (uG) H0H+H+ (uR) H−, and H0. 2) Helons bind into triplets: Helons are bound into H0H0H+ (dB ) H0H+H0 (dG) H+H0H0 (dR) triplets by a mechanism which we represent as the tops H−H−H0 (uB) H−H0H− (uG) H0H−H− (uR) of each strand being connected to each other, and the bot- H0H0H− (dB ) H0H−H0 (dG) H−H0H0 (dR). toms of each strand being similarly connected. A triplet while the leptons are: of helons may split in half, in which case a new connection + − forms at the top or bottom of each resulting triplet. The H+H+H+ (e ) H0H0H0 (νe) H−H−H− (e ) reverse process may also occur when two triplets merge Note that in this scheme we have created neutrinos, but to form one triplet, in which case the connection at the not anti-neutrinos. This has occurred because the H0 top of one triplet and the bottom of the other triplet “an- helon is its own anti-particle. This apparent problem will nihilate” each other. be turned to our advantage in Section III. Note also that The arrangement of three helons joined at the top and we have not ascribed any of the usual quantum numbers bottom is equivalent to two parallel disks connected by (such as mass or spin [13]) to the helons. It is our expec- a triplet of strands. In the simplest case, such an ar- tation that spin, mass, hypercharge and other quantities rangement is invariant under rotations through angles emerge dTheyna Topology/Stabilitymically, just like e oflec thetric Fabriccharge, as we ar- of 2π/3, making it impossible to distinguish the strands range helons into more complex patterns. without arbitrarily labelling or colouring them. However It is interesting to note that three helons seems to be we can envisage the three strands crossing over or under the minimum number from which a stable, non-trivial each other to form a braid. The three strands can then structure can be formed. By stable, we mean that a be distinguished by their relative crossings. We will ar- physical representation of a braid on three strands (e.g. gue below that braided triplets represent fermions, while made from strips of fabric) cannot in general be smoothly unbraided triplets provide the simplest way to represent deformed into a simpler structure. By contrast, such a gauge bosons. physical model with only two strands can always be un- 3) No charge mixing: When constructing braided twisted. triplets, we will not allow H+ and H− helons in the same triplet. H+ and H0 mixing, and H− and H0 mixing are Can we imagine these “particles” as bits allowed. III. PHEofN surfaceOMEN OinteractingLOGY O Fin Ta HE HELON 4) Integer charge: All unbraided triplets must carry “superficialMO spinDE Lfoam”. integer electric charge. Assumption 1) is unique to the helon model, as it reflects A braid on n strands is said to be an element of the the composite nature of helons, however assumptions 2), 3) and 4) are merely restatements in a different context of braid group Bn. Given any element of Bn, it is also pos- sible to create a corresponding anti-braid, which is the assumptions that Shupe and Harari made in their work. We will require one further assumption, however it will top-to-bottom mirror image of that braid, and is also an element of B (it is therefore merely a matter of conven- be left until later, as it can be better understood in the n proper context. tion what we call a braid or an anti-braid). If we join the strands of a braid with those of its top-to-bottom In terms of the number of fundamental objects, the helon model is more economical than even the rishon model, al- mirror image (i.e. we take the braid product [9]) we ob- tain a structure which can be deformed into the trivial beit at the cost of allowing helons to be composite. This seems a reasonable price to pay, as the tweedles are ex- braid. Conversely, if the strands of a trivial braid are crossed so as to form a braid, an anti-braid must also be

Y Twist

Figure 3: Turning over a node induces crossings and twists in the “legs” of that node (left). A trinion flip may be used to eliminate crossing while creating twists (right).

consist of n ribbons connecting one set of n points with a common z-coordinate, to another set of n points also with a common z-coordinate, as described above. In braided belts the sets of n points at both ends of the ribbon strands are replaced by disks, or framed nodes. The union of the strands and disks is a closed surface. The disks are thus nodes with n “legs” emerging from them, each leg being the terminus of a ribbon strand. We furthermore relax the condition that strands may not “loop back” on themselves. The ability to deform the braided belt by flipping the node at one end over (eectively feeding it through the strands), in order to undo crossings and hence simplify the braid structure of the belt, is essential to the results we will discuss below.

On the left of Fig. 3 we illustrate a Y-shaped ribbon - that is, a node with three ribbons or “legs” attached to it. Such a triple of ribbons meeting at a node has previously been referred to as a “trinion” [2], and we shall adopt this terminology. A trinion may be converted into a structure with both crossings and twists, by keeping the ends fixed and flipping over the node in the middle, as illustrated in Fig. 3 (we shall refer to this process of flipping over a node while keeping the ends of the legs fixed as a “trinion flip”, or “trip” for short). Conversely, on the right of Fig. 3 we show how a trinion with untwisted ribbons, but whose upper ribbons are crossed, can be converted into a trinion with uncrossed ribbons and oppositely-directed half-twists in the upper and lower ribbons by performing an appropriate trinion flip (in the illustration, a negative half-twist in the lower ribbon of the trinion and positive half-twists in the upper ribbons). In Fig. 4, we show the same process performed on a trinion whose (crossed) upper ribbons have been bent downwards to lie besides and to the left of the (initially) lower ribbon. This configuration is nothing other than a framed 3-braid corresponding to the generator 1 (with the extra detail that the tops of all three strands are joined at a node). Keeping the ends of the ribbons fixed as before and flipping over the node so as to remove the crossings now results in three unbraided (i.e. trivially braided) strands, with a positive half-twist on

5

~ ~

Twist1

-1/2 0 0 0 -1/2 -1/2 0 0 -1/2 +1/2

~ ~ ~ ~

+1/2 0

~ ~ ~ ~

Twist2

0 0 -1/2 +1/2 +1/2 0 0 0 +1/2 +1/2

~ ~ ~ ~

-1/2 0

Figure 2: The Twist

~

Figure 3: The Twist on a closed 3-Belt

4

~ ~ ~ ~

Twist ConcatenationFigure 4: Braid Makesing a Belt Braided Belts

braid B. They will get permuted in the process, according to the permutation ✏(B) associated with the braid B. Thus

⇥(B) [r, s, t]B[r⇥, s⇥, t⇥]B = [r, s, t][r⇥, s⇥, t⇥] BB⇥

where [r, s, t][x, y, z] = [r + x, s + y, t + z] and BB⇥ denotes the usual product of braid words.

For example,

[r, s, t]1[x, y, z]2 = [r, s, t][y, x, z]12 = [r + y, s + x, t + z]12

.

With the braided leather belts, we have three ribbon strands as shown in Figures 2,3 and 4. The result is three strands bound at top and bottom as shown in these figures. As the figures indicate, the braid representation formalism shows how to rewrite a braided leather belt in the form of twisted parallel strips.

1 We see from Figure 2 that [ 1/2, 1/2, 0]1 , (This is T wist1 in the Figure 2) representing three strips with frami ngs as indicated by [ 1/2, 1/2, 0] and

5

The recipe for these calculations is to write the braid in terms of the gen- erators ↵1 and ↵2 of the three-strand braid group, and add twisting the strips by using the framing vectors [a, b, c]. Then rewrite the product, as we did in the above paragraph so that it has the form [a, b, c]P where P is a product of permutation matrices. Then the twisted parallel strip framings are given by the vector [a, b, c].

In this system we represent the braided belt by using a framed braid group representation and the equivalence relation that is suggested by Figure 1. Note that in our braid group representation we have ↵ = [ 1/2, 1/2, +1/2]P and, 1 12 as in Figure 1, this corresponds to the fact the flat braid corresponding to ↵1 deforms to the belt surface with three parallel belts, each with half-twists of type [ 1/2, 1/2, 1/2]. Thus, we arrive at the follwing result. ~ ~ Theorem. To determnine the parallel twisted belt form of a framed braid product B whTheere B Positrondenotes a pandrod uItsct ofLinkedframed Boundarybraids, apply the braid group representation and reduce B to the form [a, b, c]P where P is a permuation. Then [a, b, c] is the framing on a topologically equivalent parallel strip form of the braided belt. Note that in this representation, we have only rooted the

+ bottom of the belt as shown in F~igure 1. We keep the ordere and position of the + ~ ~ top strands of the be elt fixed. ~

1 3 Example 1. The braided belt corresponding to (↵2↵1 ) is topologically equiv- alent to three untwisted parallel strips. This means that the belt with this braid- ing can be made from an untwisted belt of parallel strips by an isotopy. Our 1 3 algebraic model shows this since, as we calculated above, (↵2↵1 ) == [0, 0, 0]I

where I is the identity permutation. + ~ boundary( e ) + 1 Example 2. Let e = [1, 1, 1]↵1 ↵2. This is Sundance’s positron [2]. We find that

+ 1 e = [1, 1, 1]↵ ↵2 = [1, 1, 1][1/2, 1/2, 1/2]P12[1/2, 1/2, 1/2]P23 1 Figure 8: The Positron = [1, 1, 1][1/2, 1/2, 1/2][ 1/2, 1/2, 1/2]P P = [1, 2, 0]P P . 12 23 12 23 Thus [1, 2, 0] gives the framings on the equivalent parallel flat strip belt. See Fig- uwrelel8-dfefiornaedgrwapayh.icalIf wveersisimponlyofrtegarhis cald thcuemlatasionb,ranaiddeda bdeleptis,cttionhenofittihsepbossiounbdleartoy ofgetthdeisuerrenfacet prtohdatuctcors brespy fionrstdds eftooremi+.nWg tehdeenbelotteofthane bioundnivdiarduyalofptarhtisiclsue,rfanaced bthyenem+.ulNottipleyitnhg.at The+is imeans indeps tenhatdenthte(tfopramedologicalbrailyd) foformthuelatdefionormatis pironobabthatly wnecessare haveyapforplidedealtiongstwiraitghh tpenartouiclet itnhteerbactraiidonins,g anfromd itthleade orsigitontalhedqefiunestition of eh+ow. Thtoutshiounkrpalhgebysicalrailcyrabedouuctti”ponargitvicleses”ustanhatalhgoraveitthombeforinfianspdiecingfitchetopbouologindarcaly lfiornkmforin eacordherptaro tcomicle bonineBicorlsonrect’s ltyab. Thles.is Wisearetchualrnlentgeo thfore whtheolpehtyabsicsle atof the 1 1 ensitduatofionth.isHersecteionis an. example: u = [0, 1, 1]⌃2⌃1 and d = ⌃1⌃2 [0, 0, 1]. Thus in this form we have ud = [0, 1, 0]. However, if we reduce each of u⇥and d to Etwixampsted plear3.allelNotsteranthdats,forthenparwtieclgete inuteract[1i,on0,s1]wane wdandt w[ell1-d, efi1n,ed1].opEveridateniontlys [1, 0, 1][ 1, 1, 1] = [0, 1, 2] gives a dierent result fromth⇥e di⇥rect composition on the b⇥its ⇥of topology⇥. If we take the particles as shown in Sundance’s paper anof dthreegarframedd thembraiidns.the braid group representation, we can then multiply in a

9 4 Algebra

In this section we give a representation of the three-strand braid group that is particularly useful for understanding the properties of braided belts and the particle theory that we have discussed in the previous section. This representa- tion generalizes to the n-strand braid group, and we shall comment on this at the end of this section.

10 Apparent Moral: There is topological persistence in particle properties for the surfaces.

To what extent do the surfaces represent the elementary particles? To what extent does the framed braid and/or its algebraic representation represent the elementary particles? The Big Question: Is there simple knot theoretic combinatorial topology at the base of the world?