# Knots on a Torus: A Model of the Elementary Particles

## Abstract

**:**

## 1. Introduction

**is**an elementary particle?” Even though it evokes such matters as the meaning of the word “is”, this is not a frivolous question 1; indeed for the model to be described herein it’s fundamental. The celebrated nobelist Eugene Wigner who did so much to advance the role of symmetry in physics is credited with a detailed algebraic analysis of the subject with the conclusion that an “elementary particle “is” an irreducible unitary representation of the group, G. of physics, that is, the double (universal) cover of the Poincare group of those transformations of special relativity which can be continuously deformed to the identity” [5]. However, this is really an epistemological definition, useful for enumerating the attributes necessary for identifying a particle as elementary but providing scant guidance for describing, say, the nature of the particulate occupation of space, in some ontologically satisfying way.

## 2. The Basic Particle Model

#### 2.1. Introduction

**quirks**(not quarks!).

_{3}is the (third component of) strong isospin and Y is the so-called hypercharge which in the case of the nucleons is Y = B +1 where B is baryon number. Thus, if we equate NHT with 2I

_{3}and Q with Y/2 there is a formal equivalence between the two formulas. For example, we have with Q = e/2, NHT = −1 and +1, and q = 0 and +e, for fermions B and C, respectively (also evident from the figure) which corresponds to equating B to the neutron and C to the proton. It turns out that there is also interest in considering only the first three basic fermions (without D) in which case the average charge is 0 but the average twist becomes −1.

_{i}is the change in the ribbon’s bearing in the plane at the ith quirk and N

_{q}Is the total number of quirks. The quantum condition becomes

- ε
_{i = }+1 for ccw rotation (an “up” quirk) - = - 1 for cw rotation (a “down” quirk).

_{i}is the unit step function at ϕ

_{i}.

#### 2.2. Antiparticles and Some Aspects of Time

**Figure 15.**Figure 14 seen from the back.

**four**kinds of instanton characteristics, labeled I, AI, RI and RAI in a two-dimensional (1 + 1) spacetime as shown in Figure 16. Although the mathematical considerations are more involved, in essence, the prototypical instanton characteristic labeled I here, is just a step change between two eigenstates (which we label here as residing in the spatial dimension) that occurs in a very short interval of time, and the label AI stands for the anti-instanton characteristic with a step in the opposite direction [12]. Customarily, only the two characteristics I and AI are discussed; here, characteristics RI and RAI are introduced strictly by reasons of symmetry, RI to portray an instanton progressing in negative time and, similarly, RAI to portray an anti-instanton progressing in negative time.

## 3. Fusion

#### 3.1. Introduction

“Flattened Moebius strips (FMS)—can take the form of an elementary, triangular planform or the contiguous composite of such configurations. The composites result from an operation called fusion in which elementary configurations are combined to produce configurations with various values of twist. All values of twist can be realized in this manner but the process is degenerate; a multiplicity of configurations can exist with the same value of NHT.”

_{α}> and <Ψ

_{α}>, respectively, where the value of α is either d or u. Also, we define A and A

^{†}to be the disenabling and enabling operators4, respectively. Then, the fusing of a junction between an originally free FMS and antiFMS pair can be expressed in terms of a state function as

_{α}> to be (z

_{α}) such that z

_{d}= x and z

_{u}= y, respectively. Similarly, the operation of fission can be expressed in terms of a similar state function as

#### 3.3. Hopf Algebra

#### 3.3. First-Order Fusion

^{+ }is modeled herein, namely as a bound particle composed of a proton and an antineutron. Similarly, the one on the right represents the π

^{-}as composed of a neutron and an antiproton. Consequently, our model yields charge values of +e and −e, respectively, as it should, and twist values of 0 for both, also as it should if we associate twist with isospin as discussed in the previous discussion. The algebraic reason that each of the two pions has a vanishing twist is that the composite is the sum of two components with opposite twist in each case. The physical reason is that, although each composite can be formed by fusion (as per the above quotation) each can also be regarded as an untwisted closed band that has been subjected to an additional half twist. 5

#### 3.4. The Dirac Connection

_{α}= (ψ

_{1}, ψ

_{2})

^{T}, ψ

_{β}= (ψ

_{3}, ψ

_{4})

^{T}and

_{α}, constitutes a positive energy representation of a two-component spinor and whose lower half, ψ

_{β}, constitutes a negative energy (an antiparticle) representation of a similar spinor.

_{α }and ψ

_{β}, the two-vector halves of the four component state vector, in particular, its circular nature. That is, we see that ψ

_{α }depends on ψ

_{β }(as modified by the spin matrix) and conversely, ψ

_{β }depends on ψ

_{α}in the same manner. It turns out that this dependence implies a relationship between our particle model and the Dirac theory and to exhibit that, we begin by rewriting Equations 3-11 as

#### 3.5. Formalizing First-Order Fusion

_{0}+ V

_{1}for the case of first order fusion, that is , for s1 = s2 =1/2.

^{T}and V* = (A*, B*, C*, D*)

^{T}is the matrix of sixteen two-element composites shown in Equation 3-16. In analogy with quantum mechanics, we note for future reference that M can be viewed as an operator. Also, the equivalent matrix in standard model nomenclature is shown in Figure 40 of Section 7.4.

_{0}consists of only two elements, namely, CB* and its complement, BC* the ones discussed above and shown in Figure 20. To reiterate, although these two composites can be formed by fusion, topologically, they are really just doubled-over versions of the trivial, zero-twist MS. In other words they are excited states of the basic untwisted state and, in fact (we recall), in each case the algebraic sum of the twists of the fused constituents is zero. The other twelve bosons are all V

_{1}vector bosons in their ground state and can also be formed either by fusion or directly by a twist whose NHT is also the sum of those of its constituents.

_{2}= n – n

_{1}, we are led to the formal notion of symbolic convolution; where the direct product matrix M presents the requisite information visually, convolution does the same thing algebraically. For the case of first order fusion, we therefore define the operation of (symbolic) convolution to be

_{μ}, β

_{v}and γ

_{λ}are to be identified with the set of four basic letters, the set of four basic conjugate letters and the twist-valued headings, n, in the output of the convolution operation, respectively. That is,

## 4. Second Order Fusion

_{1/2}+ P

_{3/2}for s1 = 1 (or 0) and s2 = 1/2. The result can be viewed as a 4-vector,

_{1}is a matrix whose elements are three letter words, formed by appending the letter A to each of the elements of matrix M.

_{k}. However, since we are preoccupied with the aggregation of elements in terms of twist, we note that the twist loci are a set of inclined planes [3] where, as indicated in Figure 25 the values of twist range antiymmetrically from −9 to +9, an inclination that is quite analogous to the inclined loci of Figure 22 for the case of first order fusion and for the same reason; basically, the preservation of twist.

_{μ}, β

_{v}and γ

_{λ}are to be identified with the set of four basic letters, the set of seven columns of Figure 24 and the output of the process, respectively. An operational diagram, Figure 27, again illustrates the procedure, whose output is shown as the occupancy of the set of ten planes of Figure 28, which run from n = −9 to +9 with increments of Δn = 2, and the corresponding columns of Figure 29 shown below (by extension with the way first order assembly is organized in Figure 24).

## 5. Fusion Summary

- L = the number of basic letters (or integers) available for combination
- W = the number of letters in a word at a given OF
- S = the number of identical letters per word
- C = the number of combinations for a given set of L, W and S values.

## 6. Detailed Composition and Contingency

^{5}= 32 binary choices, twice as many as we found by considering only combinations and permutations.

AA* (x) | CA* (x) |

AB* (2x) | CB* (2x + 2y) |

AC* (x) | CC* (4y + x) |

AD* (0) | CD* (2y) |

BA* (2x) | DA* (0) |

BB* (4x + y) | DB* (y) |

BC* (2x + 2y | DC* (2y) |

BD* (y) | DD* (y) |

_{μ}and β

_{v}, are as defined in connection with Equation (3-20) of Section 3.3, coefficients ℓ and m are the number of down and up quirks in the FMS, respectively, and p and q are the number of antidown and antiup antiquirks in the conjugate FMS.

AA*A (x)(2x*) | BA*A (2x)(2x*) |

AA*B (x)(4x*) | BA*B (2x)(4x*) |

AA*C (x)(2x*) | BA*C (2x)(2x*) |

[AA*D] | [BA*D] |

AB*A (2x)(x*) | BB*A {(4x)(x*) + (y)(4x*} |

AB*B (2x)(2x* + y*) | BB*B {(4x)(2x* +y*) + (y)(4x*} |

AB*C (2x)(x* + 2y*) | BB*C {(4x)(x* + 2y*) + (y)(2y*)} |

AB*D (2x)(y*) | BB*D (4x)(y*) |

[AC*A] | BC*A (2y)(x*) |

AC*B (x)(2y*) | BC*B {(2x)(2y*) + (2y)(2x* + y*)} |

AC*C (x)(4y*) | BC*C {(2x)(x* + 2y*) + (2y)(x* + 2y*)} |

AC*D (x)(2y*) | BC*D {(2x)(2y*) + (2y)(y*)} |

[AD*A] | [BD*A] |

[AD*B] | BD*B (y)(2y*) |

[AD*C] | BD*C (y)(4y*) |

[AD*D] | BD*D (y)(2y*) |

## 7. The Standard Model Connection

#### 7.1. Ambiguities

#### 7.2. Delta Creation and Decay

^{0R}and π

^{0L}, shown here of the neutral pion rather than the one, π

^{0}usually shown in the SM.

^{+}and π

^{-}, respectively, note that if n is equated with B and p with C, then, as per this matrix (recall that fermion twist increases to the right and antifermion twist upward) we see that

^{+}and Δ

^{-}are two members of the last triplet in the NHT = −1 column of that figure and Δ

^{++}and Δ

^{0}are two members of the last triplet in the NHT = +1 column. However we can also express two of the delta particles as

^{0}.” In the same vein, we note that the two pion versions can also form the superposition for the η particle, viz:

#### 7.3. Beta Decay Examples

_{e}(for its lepton number).

^{- }particle (see the boson matrix M, Equation (3-16)) which then splits into an electron (i.e., A) and its antineutrino (i.e., B*)

_{e}in order to cancel the electron’s lepton number. And finally, the B of the second fermion must translate to a muon neutrino v

_{μ}in order to match lepton numbers, whereupon we end up with AB*B (see Figure 41) which translates into W

^{-}v

_{μ}which splits up into an electron and its antineutrino and the muon’s neutrino.

#### 7.4. The Boson Matrix as per the SM

^{-}, W

^{+}and Z

^{0}, are in the lower left quadrant and in the (diagonally) next, overlapping quadrant are the pions involved in “strong” interactions: π

^{-}, π

^{+}, π

^{0L}and π

^{0R}. In the SM these two groups of operators enact parallel roles in terms of weak and strong isospin, respectively. That is, the manner in which the pions in the second quadrant act upon the nucleons is identical to the manner in which the vector bosons in the first quadrant act upon the electron/(anti)neutrino pair of leptons. In [1] a model, reproduced below in Figure 43, shows pions mediating Yukawa type exchanges between nucleons to maintain Deuteron stability in what may be characterized as strong isospin space.

^{-}splits into a free neutron and an antiproton which fuses with the original proton to make the π

^{0R}and, similarly, the π

^{+}splits into a free proton and an antineutron which fuses with the neutron to form the π

^{0L}. That is we have the interactions

^{+}splitting in analogy with the π

^{-}and so on to perpetuate the electron/antineutrino pair. That is, we begin by making the correspondences:

^{0}.

## 8. Expansion to Three Generations

#### 8.1. Scaffolds for the Family Tree

#### 8.2. Triplication of Time

#### 8.3. Occupancy of the Three Generational Scaffold

^{0}and Λ

^{0}particles, with the overall result that we realize, as per [1], the SM’s spin ½ baryon octet.

## 9. Some Quantum Mechanical Connections

^{-}= BC*, π

^{0L}= BB*, π

^{0R}= CC*, π

^{+}= CB*.

_{1}+ n

_{2}. Clearly, we have commutation only for n

_{1}= n

_{2}(or x = y) which translates to bosons located along the twist gradient of matrix M (or Figure 22).

## 10. Retrospective

- 1. The particles of the model are localized distortions in and of an otherwise undistorted spacetime.
- 2. There is an underlying toroidal topology embodied in a basic set of four Moebius strips (MS), or equivalently, two rudimentary torus knots, each twisted both left and right.
- 3. To create a basic set of particles, each of the four MS is flattened into a triangular two dimensional planform, an FMS.
- 4. A counterclockwise direction of traverse is chosen around the planforms.
- 5. Time is identified as the odd dimension in the 2 + 1 dimensions occupied by each FMS
- 6. A corresponding basic set of four conjugate particles (antiparticles) is defined by a reversed direction of traverse.
- 7. All other particles are constructed as composites by fusing basic particles and antiparticles.

## 11. Some Connections

#### 11.1. String Theory

#### 11.2. TQFT

#### 11.3. New Particles

^{+}meson (http://www.phy.ohiou.edu/~hicks/thplus/thplus3.html).

^{+}meson. As pointed out in [1] this sounds very much like our delta particle

**model**(specifically a delta plus) if we also recall the transformation of a π

^{+}into a K

^{+}by the change of a d* antiquirk into an s* antiquirk as portrayed in Figure 54 of Section 8.

^{+}which tallies with the figure. Also found were a proton and a K

^{-},which as in [1], suggests the delta minus-like particle shown below but with a π

^{-}that has similarly decayed into a K

^{-}.

#### 11.4. Dark Matter

## Appendix A. Solitonic Behavior

^{2}+ μ

^{2}r

^{2})]

^{1/2}, which, is to first order in r/R.

^{4}/4πG, kg (m/s)

^{2}/m, c is the velocity of light and G ≌ 6.673 × 10

^{-11}m

^{3}/kg⋅s

^{2}is the gravitational constant. A has the dimensions of energy per unit length and, correspondingly, B is energy per unit volume but is unspecified at this point. The integration is over some putative volume enclosing the knot.

^{μk}R

_{μk}is the Curvature Scalar (CvS) and

_{μv}, the Energy-Momentum or Stress Energy Tensor (EMT),

_{L}(and so forth for the derivatives). We then begin as in the above with a corresponding replacement Equation (A-17) namely

_{R}is the same but a

_{L}and F

_{L}both now incorporate θ

_{L}rather than θ. Thus, to recreate the format of the dynamic equation we must define the mass term for leftward traverse as m

_{L}= -m

_{R}. In other words, the mass associated with traverse to the left is the negative of the mass associated with traverse to the right.

## Appendix B. Contingency in Second Order Fusion

AA*A (x)(2x*) | BA*A (2x)(2x*) |

AA*B (x)(4x*) | BA*B (2x)(4x*) |

AA*C (x)(2x*) | BA*C (2x)(2x*) |

[AA*D] | [BA*D] |

AB*A (2x)(x*) | BB*A {(4x)(x*) + (y)(4x*} |

AB*B (2x)(2x* + y*) | BB*B {(4x)(2x* +y*) + (y)(4x*} |

AB*C (2x)(x* + 2y*) | BB*C {(4x)(x* + 2y*) + (y)(2y*)} |

AB*D (2x)(y*) | BB*D (4x)(y*) |

[AC*A] | BC*A (2y)(x*) |

AC*B (x)(2y*) | BC*B {(2x)(2y*) + (2y)(2x* + y*)} |

AC*C (x)(4y*) | BC*C {(2x)(x* + 2y*) + (2y)(x* + 2y*)} |

AC*D (x)(2y*) | BC*D {(2x)(2y*) + (2y)(y*)} |

[AD*A] | [BD*A] |

[AD*B] | BD*B (y)(2y*) |

[AD*C] | BD*C (y)(4y*) |

[AD*D] | BD*D (y)(2y*) |

Available Junctions | Available Antiquirks | Available Junctions | Available Antiquirks |
---|---|---|---|

AA*(x) | 2d* | CA*(x) | 2d* |

AB* (2x) | 2(d* + u*) | CB*(2x + 2y) | 4d* + 2(d* + u*) |

AC* (x) | 2u* | CC*(4y + x) | 4(d* + u*) + 2u* |

AD* (0) | 0 | CD*(2y) | 2u* |

BA* (2x) | 2d* | DA*(0) | 0 |

BB* (4x +y) | 4(d* +u*) + 2d* | DB*(y) | 2d* |

BC* (2x + 2y) | 2(d* + u*) + 4u* | DC*(2y) | 2d* + 2u* |

BD* (y) | 2u* | DD*(y) | 2u* |

_{μ}β

_{v}(rx + sy) meaning that (r) FMS with (x) type junctions and (s) FMS with (y) type junctions can be formed in each case. The logic displayed in the example then translates into an expression for an enhanced product with the antiquirks available for fusion in the form

## Appendix C. Hopf Algebra

^{2}. For the special case of q = −1, these become

^{-}and W

^{+}) and the pair (BC* and CB*) which correspond to the charged pions. In summary, given the identification the model’s fusion and fission operations with the algebra’s multiplication and comultiplication operations plus the identification of bound particle conjugation in the model with the algebraic concept of an antipode, we conclude that our model qualifies as a Hopf algebra, a conclusion that awaits formal documentation.

## Appendix D. The Beta Switch

## Appendix E. Quaternions

#### E.1. Introductory

_{0}or q

_{0}in Figure E1 we are left with a 3 × 3 matrix, the direct product of the 3-vectors or alternatively, the direct sum of the inner product and cross product of the vectors shown in Figure E2. We also note that, from the quaternion point of view, vectors are also known as pure quaternions and the 3 × 3 matrix is a 2-dimensional manifestation of their outer product. In any event, whether in full or reduced form, the matrix of Equation (E-3) constitutes an operational representation of the outer product of two quaternions, and can be used to operate on additional quaternions.

#### E.2. Time Triplication

_{i}with u, c, and t, respectively and j = 1, 2, 3 of the t

_{j}

^{*}with d, s, and b, respectively, we see that the two matrices are isomorphic. Also, as noted in [17] a useful approximation to the KM matrix is

_{12}, and s

_{ij}= sinθ

_{ij}, the angle being the “mixing angle” between the ith and the jth generations. Thus our t

_{1}t

_{2}* and t

_{2}t

_{1}* terms correspond to the KM terms λ and -λ, respectively, of the first and second generations, as they should, and our t

_{2}t

_{3}* and t

_{3}t

_{2}* terms correspond to the KM terms λ

^{2}and –λ

^{2}of the second and third generations, as they should.

#### E.3. Taxonomical Implications

**Figure E4.**Cross product and scalar components of Figure E3.

## Appendix F. On the Kauffman Bracket Polynomial of (2, n) Torus Knots

^{-3}(which is equal to K

_{-1}) and y ≡ A and K

_{0 }= -(A

^{2}+ A

^{-2}).

_{0}input exists for only an initial sample). Due to the circulation around the closed loop, the system response at time n to an impulsive input occurring at the previous time, m, is y

^{u-m}(or A

^{n-m}). Furthermore, by time n, there are n such responses to be added up, thus implying the summation in Equation (F-2). Also, since there is only a single initial impulse, K

_{0}, switched on at m = 0, the associated response (the result of summation) at time n is just A

^{n}K

_{0}. Note that the output shown in the figure is just Equation (F-3).

_{1}+ n

_{2}. We find that

_{1}+ n

_{2}+ n

_{3}, proceeds in essentially the same way and in analogy with Equation (F-5) we find

_{i}= α

^{ni}, α = x, y, i = 1, 2, and 3, , and the convolutions, C

_{i}, are as before but with i = 1, 2, and 3.

_{1}= y

_{2}y

_{3}+ y

_{2}x

_{3}+ x

_{2}x

_{3}, β

_{2}= y

_{3}y

_{1}+ y

_{3}x

_{1}+ x

_{3}x

_{1}, β

_{3}= y

_{1}y

_{2}+ y

_{1}x

_{2}+ x

_{1}x

_{2}.

_{F}, of components. In analogy with Equation (F-8) we find that

_{F}, the coefficients (the γ's) are computable in the manner illustrated for the case of n

_{F}= 3. However, we can anticipate the results on the basis of the following selection rules:

- 1. All subscripts, including that of the C's, must be present in each coefficient.
- 2. The number of y terms decreases linearly from n
_{F}-1 in the first coefficient to 0 for the last, while the number of x terms increases, correspondingly, from 0 in the first coefficient to n_{F}-1 in the last. - 3. The highest y subscript value is n
_{F}and the highest x subscript value is n_{F}- 1.

_{F}= 4 as

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^{1 }Not that the meaning of the word "is" is frivolous!.^{2 }The search for the Higgs goes on.^{3 }More on flattening below.^{4 }* Note the analogy to the annihilation and creation operators of quantum field theory^{5 }which may have relevance to the subject of particulate genesis (but not in this paper).^{6 }Note the analogy to the SM's (enigmatic) switch of the neutron’s down quark to an up quark, by means of which the neutron is converted into a proton plus a W particle.^{7 }We note the correspondence to the six-dimensional vector space (three electric and three magnetic} of electromagnetism.^{8 }Emphasis added^{9 }Not a new result but presented as emerging from a unique basis.^{10 }Computational details are provided in [4].

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Avrin, J.S.
Knots on a Torus: A Model of the Elementary Particles. *Symmetry* **2012**, *4*, 39-115.
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Avrin JS.
Knots on a Torus: A Model of the Elementary Particles. *Symmetry*. 2012; 4(1):39-115.
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Avrin, Jack S.
2012. "Knots on a Torus: A Model of the Elementary Particles" *Symmetry* 4, no. 1: 39-115.
https://doi.org/10.3390/sym4010039