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On the relation between the three Reidemeister moves

and the three gauge groups

Christoph SchilleraID

2023

Abstract

Quantum theory suggests that the three observed gauge groups U(1), SU(2) and SU(3) are

related to the three Reidemeister moves: twists, pokes and slides. The background for the

relation is provided. It is then shown that twists generate the group U(1), whereas pokes

generate SU(2). Emphasis is placed on proving the relation between slides, the Gell-Mann

matrices, and the Lie group SU(3). Consequences for uniﬁcation are deduced.

Keywords: Reidemeister moves; gauge groups.

aMotion Mountain Research, 81827 Munich, Germany, cs@motionmountain.net, ORCID 0000-0002-8188-6282.

2

Reidemeister move I,

or twist

Reidemeister move II,

or poke

Reidemeister move III,

or slide

Fig. 1 Three moves – actually shape deformations – for links, knots, tangles and braids were

deﬁned by Reidemeister [1]: twists, pokes and slides. In the strand tangle model of quantum

theory, they maintain the topology and thus maintain the particle type, but they change the

phase of the quantum state. Therefore they model interactions.

1 The origin of the gauge groups

Clarifying the origin of the three gauge groups for the three gauge interactions is an open issue of

physics. Often, the three gauge groups U(1), SU(2) and SU(3) are seen as related to the complex

numbers, the quaternions and the octonions [2–4]. However, this approach, while successful for

U(1) and SU(2), yields SU(3) only indirectly, and only together with additional structures. In the

following, it is argued that a simpler way to understand the origin of the gauge groups is by relating

them to the Reidemeister moves.

The three Reidemeister moves, illustrated in Figure 1, have been used to classify the possible

deformations of mathematical links, knots, tangles and braids for almost a century. The relation

between Reidemeister moves and gauge groups that is used in the following arose from Dirac’s de-

scription of spin 1/2particles as tethered structures. In his lectures, Dirac popularized a variation

of the belt trick – without ever publishing anything about it – that shows how tethered structures

behave like spin 1/2particles: they come back to their original state only after a rotation by 4π, i.e.,

after two full turns [5]. Tethered structures also behave as fermions under exchange. Continuing

this approach, Battey-Pratt and Racey were led to explore tethered particles and the behaviour of

their tethers more thoroughly, and to associate tethers to wave functions. This association allowed

them to deduce the Dirac equation [6]. Extending this approach led to the so-called strand tangle

model, which describes elementary particles as rational tangles of strands and gauge interactions

as tangle deformations [7–10]. The present text focuses on how tangle deformations are related

to the generators of nature’s gauge groups. It will be shown that each of the three Reidemeister

moves is related to one of the three observed gauge groups which describe the gauge interactions

3

Strand description:

The fundamental Planck-scale principle of the strand tangle model

Resulting

observation:

A localized

fundamental

event

t t + ∆t

W=~

∆l≥p4~G/c3

∆t≥p4~G/c5

S=kln 2

Fig. 2 In the strand tangle model, the only observable process is the local change of sign of a

strand crossing, the so-called crossing switch. This fundamental process generates a quantum

of action ~. The strands themselves are of Planck radius and thus are not observable. A

crossing is a region where the strand distance is minimal. Only a crossing switch is observable.

As shown in previous papers [7–12], this fundamental principle yields the Lagrangian of the

standard model of particle physics, extended with massive Dirac neutrinos and PMNS mixing,

and the Lagrangian of general relativity.

between fermions.

2 A summary of the strand tangle model

The present section presents the physical background. In nature, gauge interactions change the

state of fermions: gauge interactions change the phase of particle wave functions. To explore the

relation between gauge groups and Reidemeister moves, it is thus necessary to clarify how strands

lead to particles, to their phase, and to wave functions. This was done extensively by Battey-Pratt

and Racey [6] and in previous articles [7–10], all based on Dirac’s explanation of spin 1/2particles

using tethered structures.

In the strand tangle model, particles are rational tangles of strands, as illustrated in Figure 3.

Strands are ﬂuctuating lines – actually of Planck radius – in space, with no ends, that reach the

cosmological horizon. Because of the minimum length in nature, strands are impenetrable, uncut-

table and unobservable. The only observable process occurs when, in a region in space, a strand

segment that was behind another strand, suddenly is in front of the same strand. This local process

– which slightly depends on the observer – is called a crossing switch. In the strand tangle model,

a crossing switch generates a quantum of action ~. This statement, illustrated in Figure 2, is the

fundamental principle of the strand tangle model. Every crossing switch is observable, and every

observable is due to crossing switches.

All particles are rational tangles of strands. Therefore, all particles are tethered structures.

The tethers are those strand segments that reach the cosmological horizon. The tangle topology

4

The strand tangle model for a fermion in the Schrödinger picture:

tangle

core

tether

Step 3:

The time average

of crossing switches

y y

dy

.

Observation:

probability

density

crossing

midpoint

spin

orientation

Crossing midpoints

with their amplitudes

and phases

Step 1:

Ody

the

crossing midpoints

of the tangle and

their phases are

d.

Step 2: The time averages

of all crossing midpoints

and of all phases

y

t e

dd.

Wave function:

amplitude and

total phase

total

phase

Spinning

fermion tangle

The shortest distance s determines the

amplitude, black dot at midpoint speci

es

the crossing position, and the angle

de

nes the main phase

.

crossing

axis

phase

s

1

2

,

Fig. 3 In the strand tangle model, ﬂuctuating strands of a particle tangle yield wave functions

and probability densities. Tangles are ﬂuctuating skeletons of wave functions.

determines the type of the particle – i.e., the substance. For example, the conservation of topology

over time reproduces the conservation of particle type. The average geometrical shape – i.e., the

average form – determines the state of the particle. In particular, the (average) location of the

tangled part, the tangle core, determines the position of the particle. The phase of a wave function

is the (average) orientation of the tangle core. The wave function arises through the blurring (i.e.,

the averaging) of the ﬂuctuating tangle crossings. Conversely, the tangle is the ﬂuctuating skeleton

of a wave function. The derivation of the probability density from a tangle is brieﬂy illustrated in

Figure 3. Spinors and antiparticles can also be deduced.

Particles are rational tangles of strands. Rational tangles are unknotted tangles that can be

untied by moving the tethers around. Braids are typical examples of rational tangles. Elementary

particles are rational tangles consisting of one, two or three strands. Classifying such rational

5

Electromagnetic interaction is twist transfer

fermion

virtual

photon

Weak interaction is poke transfer

fermion

virtual

weak

boson

Strong interaction is slide transfer

fermion

virtual

gluon

vacuum

vacuum

vacuum

Reidemeister move I

or twist

Reidemeister move II

or poke

Reidemeister move III

or slide

twists have

one generator

that generates

U(1)

pokes have

3 generators

that generate

SU(2)

slides have

9 - 1 = 8

generators

that generate

SU(3)

Fig. 4 The strand tangle model of gauge interactions is based on the theorem that the three

Reidemeister moves classify all possible deformations of tangle cores [1]. In particular, the

Reidemeister moves determine the generators and the elements of the three observed gauge

groups, as shown below. A ﬁrst, simpliﬁed way to describe the Reidemeister moves is by

stating that they rotate the segment enclosed by a dotted circle by the angle π– and then shift

it, as shown later on. This model yields quantum ﬁeld theory, as shown in references [7–10].

tangles leads to the observed spectrum and quantum numbers of elementary particles [7–10].

In the strand tangle model, gauge interactions are tangle core deformations. Core deformations

lead to a phase change for the particle. About a century ago, Reidemeister proved that all core

deformations that maintain tangle topology can be composed of just three basic types, illustrated

in Figure 1, which he called moves [1]. The general relation between the three moves and the

gauge interactions is summarized in Figure 4. The core deformation process leads to a model for

the emission and absorption of gauge bosons; in turn, this process allows reproducing perturbative

6

A double twist of the encircled segment can be rearranged t

o

ed strand,

keeping the encircled segment

x

ed in space: no

I

ermion tangle core,

o

en

x

thus generates a U(1) Lie group.

The twist, or

rst Reidemeister move, is related t

o !o

!c

T

w

performed b

"

rotating the encircled segment, are thus observable

c

#

#

x

#

rst

second

rearrangement no

Fig. 5 Top: Twists, the ﬁrst Reidemeister moves, can be described as the local rotation by πof

an encircled (red) strand segment around a given (dotted) axis. The dotted circle can be

imagined to be the border of a transparent plastic disc glued to the segment, or a ﬁst from the

left grabbing the segment. Bottom: A double twist, a full turn of the (red) segment, is

topologically equivalent, modulo strand rearrangements, to no twist at all. As a consequence,

as shown in the text, twists generate the Lie group U(1).

quantum ﬁeld theory [9,10].

In short, tangles of unobservable strands with observable crossing switches reproduce spin 1/2,

wave functions, phase, probability densities, spinors, and quantum (ﬁeld) theory. As will be shown

now, the strand tangle model reproduces the three gauge groups.

3 U(1) and the ﬁrst Reidemeister move

The ﬁrst type of fermion core deformations of interest in quantum theory is the ﬁrst Reidemeister

move, the (full) twist, which is illustrated in Figure 5. In three dimensions, a (full) twist can be

described as a rotation of a strand segment by πaround a given axis. If desired, one can imagine

that the dotted circle in Figure 5is a ﬁst grabbing the segment from the left, or that it is the border

of a transparent plastic disc on which the strand segment is glued. Such a localized strand segment

rotation by πbehaves like the generator of U(1), as shown in the following. The arguments about

twists are a warm-up for the cases of pokes and slides.

In three dimensions, a crossing is an observer-dependent feature. For a small set of observers,

7

located in the plane deﬁned by the straight strand and the rotation axis, the rotation of a strand

segment by πaround the axis does not generate a crossing, and thus does not generate a twist. Now,

the basis of the strand tangle model is that only crossing switches are observable, i.e., only the

disappearance of a crossing with one sign and the appearance of a crossing with the opposite sign

are observable. This implies that the mentioned small set of observers that do not observe crossings

make no observation at all. In addition, due to the continuous ﬂuctuations of strand shapes, the

small set of observers is negligibly small; it thus can safely be ignored. The overwhelming majority

of observers will observe a crossing when a twist move is performed. And as a consequence of

the fundamental principle of the strand tangle model, twists, or ﬁrst Reidemeister moves, are

observable. And indeed, every twist on a fermion core yields a change in its phase.

The twist illustrated in Figure 5can be generalized to arbitrary angles: one can imagine that the

strand segment enclosed by the dotted circle – or, if desired, the transparent plastic disc containing

it – is rotated by an arbitrary angle around the rotation axis. (Sometimes an additional shift of the

segment perpendicularly to the tethers is helpful for visualization.) Such a local deformation is

best called a generalized twist. (Generalized twists do not produce crossings for all observers.)

Generalized twists on a given strand segment around a given axis obey the group axioms: they

can be concatenated (multiplied), the concatenation is associative, there is a neutral element (no

rotation at all), and every generalized twist has an inverse (the inverse rotation of the segment).

Thus, generalized twists form a group. Generalized twists are parametrized by a single real angle:

they form a manifold. Concatenations behave nicely on this manifold. Therefore, generalized

twists form a one-dimensional Lie group. The concatenation of generalized twists is also com-

mutative. Above all, as shown in Figure 5, the concatenation of two (full) twists is topologically

equivalent, modulo strand rearrangements, to no twist at all. Full twists thus behave like rotations

by πon a circle: two such rotations are equivalent, modulo 2π, to no rotation. The group deﬁned

by generalized twists is thus compact. The only one-dimensional, commutative, and compact Lie

group is the circle group U(1), also denoted S1. U(1) is also the set of unit complex numbers.

Recapitulating, the description of a fermion with a tangle implies that gauge interactions are

due to strand deformations and change the total phase of the fermion. Twists are a class of strand

deformations and form a group. The ﬁliform structure of strands implies that the representations

of their deformation group are unitary. Modelling gauge interactions with deformations of strands

thus explains the ‘U’ of U(1). The single strand involved in twists explains the ‘(1)’ of U(1).

In short, generalized twists for a given strand segment around a given axis form the Lie group

U(1). The ﬁrst Reidemeister move, the (full) twist, acts as the generator of the group U(1) of

generalized twists. In the strand description of wave functions, particles and interactions, the twist

plays an important role. A twist changes the phase of a fermion and thus models an interaction. A

separate publication has shown that Figure 4and Figure 5can be used to deﬁne a complete model

for the electromagnetic interaction, for the electric charge, for the photon, for the electromagnetic

8

Pokes, or second Reidemeister moves, on pairs of strand segments form an SU(2) Lie group,

because the three rotations by reproduce the SU(2) algebra of the belt trick:

Fig. 6 Top: the three types of (full) pokes – second Reidemeister moves – are illustrated.

Pokes are most practically described by rotating by πaround the rotation axis a dotted region –

or a transparent plastic disc – containing two strand segments. The middle image illustrates the

textbook form of the poke. The pokes for the other two rotation axes are illustrated on the

outer sides. Bottom: fourfold pokes – i.e., two turns of the plastic disc (or belt buckle) – are

topologically equivalent, modulo strand rearrangements, to no poke at all. This is the belt trick.

As a result, as shown in the text, pokes generate the Lie group SU(2).

coupling constant, and for perturbative quantum electrodynamics [9]. Twists in strands predict

that no measurable deviation from quantum electrodynamics will ever be observed.

9

4 SU(2) and the second Reidemeister move

The next type of fermion core deformation of interest in quantum theory is the second Reidemeister

move, the (full) poke, which is a local rotation of a region enclosing two strand segments by π,

as illustrated in Figure 6. Three linear independent local rotations are possible, along the three

axes that are perpendicular to each other. The three pokes are called τx,τyand τz. One can

visualize each poke with the rotation of a transparent disc containing the two strand segments. The

transparent disc behaves like a belt buckle. These three types of pokes behave like the generators

of SU(2), as shown in the following. The arguments are well-known from the belt trick – also

called the scissor trick, plate trick, Dirac trick, or quaternion demonstrator. The arguments are

given in a way that is also helpful for the next section, which will explore slides.

The basis of the strand tangle model is that only crossing switches are observable, i.e., only

exchanges of a crossing with a crossing of opposite sign are observable. It is worth recalling

that in three dimensions, crossings are (slightly) observer-dependent features. A very small set of

observers of the deformations illustrated in Figure 6, located in a plane whose location depends on

the speciﬁc rotation by π, observe no crossing. Again, due to the continuous ﬂuctuations of strand

shapes, the very small set of observers can be neglected. The overwhelming majority of observers

will observe crossings when a poke is performed. As a consequence of the fundamental principle

of the strand tangle model, pokes, or second Reidemeister moves, are observable. Speciﬁcally,

every poke in a particle core yields a change in the phase of the particle.

The mathematical properties of pokes can be explored with an analogy from the human body.

One’s hand can be taken as the circled region, and the arm represents three or more tethers. If

desired, one can take two hands holding each other, to represent even more tethers. Exploring

the behaviour of the tethered circled region – or transparent plastic disc, or belt buckle, or hand –

deﬁned in Figure 6, one ﬁnds several results. First, concatenating two pokes by the angle πaround

two perpendicular axes yields the third – or its negative. Second, the concatenation of different

pokes around two perpendicular axes anti-commutes.

Finally, one ﬁnds that for all three pokes, the fourth power is the identity. This result, il-

lustrated in Figure 6, is the belt trick used by Dirac. The belt trick shows that a tethered

object – such as a hand, a belt buckle or the plastic disc in the ﬁgure enclosing two strand

segments – returns to its previous situation every two turns. An animation of the belt trick

by Antonio Martos and animations for various numbers of tethers by Jason Hise are found at

https://www.motionmountain.net/videos.html#strands. The fourth power of each poke, a rotation

by 4π, is the identity, and the square of each poke, a rotation by 2π, is −1. This applies to any

belt buckle or structure with more than two tethers (or with at least one belt). All properties can

10

be summarized in the following concatenation (multiplication) table for the generators

·τxτyτz

τx−1−τzτy

τyτz−1−τx

τz−τyτx−1

(1)

In this table, the entry −1means that the belt buckle (or the dotted circle, or the hand) has rotated

by 2πand that the strand tethers (or the arm) are twisted. One notes that the operators τnbehave

like itimes the Pauli matrices:

τx=iσx=i 0 1

1 0!, τy=iσy=i 0−i

i0!, τz=iσz=i 1 0

0−1!(2)

Using the relation that icorresponds to a rotation by π, one can deduce the matrix entries in the

poke matrices directly from the behaviour of the two strand segments illustrated in Figure 6: each

entry tells whether an encircled segment is rotated or not, whether it switched position with the

other one or not, and how the segments switched.

In total, all the properties deduced from Figure 6prove that the three pokes generate the Lie

algebra of SU(2). If one prefers, one can deduce the Lie algebra of SU(2) also by introducing the

commutator between pokes. For example, this can be done by using the matrix representations.

The step from the Lie algebra to the Lie group arises by generalizing the three pokes to arbitrary

angles: one can imagine that the dotted circles containing the two segments in Figure 6are rotated

by an arbitrary angle around the rotation axes. Such deformations are best called generalized

pokes. (Also generalized pokes produce crossings only for some observers.)

Generalized pokes obey the group axioms: they can be concatenated (multiplied), the concate-

nation is associative, there is a neutral element (no rotation at all), and each generalized poke has an

inverse (the inverse rotation of the segments). Generalized pokes thus form a group. Generalized

pokes are parametrized by three real angles, they form a manifold, and their multiplication behaves

nicely on this manifold. Therefore, generalized pokes form a three-dimensional Lie group. Being

described by angles, the Lie group deﬁned by generalized pokes is also compact. Together with

the Lie algebra of the generators, this implies: generalized pokes form the Lie group SU(2). The

Lie group SU(2) is also the set of unit quaternions.

The description with tether deformations, the unitarity of the group elements, and the Hermitian

property of the Pauli matrices all imply each other. Modelling gauge interactions with strand de-

formations thus explains the ‘U’ of SU(2). Without the tethers, unitarity would not arise. Without

tethers, the group arising from the rotations of the dotted circle would be orthogonal, not unitary,

and would be SO(3). The impenetrability of strands implies vanishing traces of the representing

11

matrices, implies the determinant +1, and explains the ‘S’ of SU(2). Finally, the two strands

involved in pokes explain the ‘(2)’ of SU(2).

It should be remarked that generalized pokes can also be described by the deformations of a

single strand segment – i.e., with a plastic disc containing just one strand segment – provided that

the deformations with respect to the other strand are deﬁned properly. One then needs to combine

arotation of the disc with a linked translation with respect to the other strand: this is a screw-like

motion of the strand segment. (In a screw, rotation and translation are linked.) The results for

single strand deformations are the same as those just derived for strand pair deformations. The

description using deformations of strand pairs was chosen above because the analogy with the belt

trick is more intuitive, and because it is helpful in the next section.

In short, the second Reidemeister move, the poke, yields three deformations that generate the

Lie algebra SU(2). This is as expected and known from the belt trick. Generalized pokes form the

full Lie group SU(2). In the strand description of wave functions, particles and interactions, pokes

play an important role. A poke changes the phase of a fermion and thus models an interaction.

Pokes and their Lie group SU(2) can be used to deﬁne a model for the weak interaction and

for the weak bosons (before symmetry breaking). Also parity violation, the mixing with quantum

electrodynamics, symmetry breaking, and all other effects of the weak interaction arise naturally in

the strand tangle model, including the weak coupling constant, the mixing angles and the quantum

ﬁeld theory of the weak interaction [7,8]. Pokes in strands predict that no deviation from the usual

description of the weak interaction – called quantum asthenodynamics by Weisskopf – will ever

be observed.

12

Slides, or third Reidemeister moves, can be described by deformations of one strand or o

f $%& '$()*+',

H-(- /' $0- $0/(+ 1

eidemeister move for one, black

'$()*+, 2$ (&$)$

es and shifts the dotted disc:

2*'$

ead of rotating and shifting a segment of the black strand, one can rotate and shift the crossing

segments of the other two

'$()*+', H-(- )(- $0- 3-*-()$

ors of the ﬁrst SU(2) subgroup of SU(3):

Fig. 7 Three types of the (full) third Reidemeister moves – three types of slides – are

illustrated. In this triplet, each move deforms a pair of crossing strands and leaves one (black)

strand undeformed. Each move can be seen as a rotation of the region inside the dotted circle

plus a related shift of that circle in the direction of the central starting triangle, in the way of

the motion of a screw. The three moves generate an SU(2) (sub)algebra that corresponds to the

belt trick for the two crossing strands. Two further triplets of slides are illustrated in Figure 8.

5 SU(3) and the third Reidemeister move

The last type of fermion core deformations of interest in quantum theory is the third Reidemeister

move, the (full) slide, which occurs in conﬁgurations in which three strands are on top of each

other, as illustrated in Figure 7. In particular, a slide can be described as a deformation of a (dotted

circle) region of two crossing strand segments against a third strand: the deformation consists of

a rotation of the region by πand a subsequent linked shift towards the third strand. Rotations and

shifts are linked as in the motion of a screw. Three such local rotations of strand pairs by πand

linked shifts are illustrated in the ﬁgure.

The basis of the strand tangle model is that only crossing switches are observable, i.e., only the

disappearance of a crossing and the subsequent appearance of a crossing with the opposite sign

13

Fig. 8 The ten important deformations deduced from the third Reidemeister move, the slide,

are illustrated. (The graphs are squashed vertically to save space.) Each deformation, apart

from λ8, rotates a dotted circle by π. Each of the upper three rows deﬁnes an SU(2) subgroup.

Using the deﬁnition of λ8given in the text, the eight slides iλ1,...,iλ8, thus without iλ9and

iλ10, turn out to generate the Lie group SU(3). The corresponding Gell-Mann matrices are

given in Table 1.

14

are observable. As a consequence of the fundamental principle of the strand tangle model, also

third Reidemeister moves, or slides, are observable. Every slide yields a change in the phase of

the fermion tangle.

The three deformations of Figure 7are linearly independent, as they take place along mutually

perpendicular axes. Imagining a transparent plastic disc glued to the crossing strands inside the

dotted circle helps to visualize the deformations. Like in the situation described in the previous

section, on pokes, the circled region behaves like the belt buckle in the belt trick. The three

deformations of Figure 7thus act as generators of an SU(2) subalgebra.

In a situation with three strands, for each strand pair there are three different rotation-shifts.

This yields a total of nine deformations. The nine deformations are illustrated in the upper three

rows of Figure 8. All deformations in the ﬁgure are called full slides in the following. As shown

next, the nine types of deformations just deﬁned can be combined into eight linearly independent

ones that behave like the generators of SU(3). This is most simply done by deﬁning an additional

deformation that is illustrated at the bottom of Figure 8, bringing the total to ten slides. To show

that SU(3) arises from Figure 8, the Gell-Mann matrix representation of Table 1and the multipli-

cation Table 2are deduced in the following. They imply the Lie algebra of SU(3). Afterwards, the

Lie group is deduced.

Figure 8implies that the slides λ3,λ9, and λ10 are linearly dependent. Indeed, these three

deformations act in related ways, in the same plane, on the three strands. The ﬁgure shows that

only two of three slides λ3,λ9and λ10 are linearly independent: two types of slides are sufﬁcient to

move all three strands. To have an orthonormal basis, it is customary to use the deformation λ3and

the additionally deﬁned deformation λ8= (λ10 −λ9)/√3. The factor √3 = 2 sin(2π/3) is due

to the angle 2π/3that describes the threefold axis at the centre of the three-strand conﬁguration.

As a consequence of this threefold symmetry, the square root of three appears in many places in

SU(3).

Using the deﬁnition of iλ8, the eight slides iλ1,...,iλ8illustrated in Figure 8are all linearly

independent of each other. In particular, the ﬁgure illustrates that iλ1,...,iλ7either act on differ-

ent dotted circles or at least act along linearly independent directions. The generator iλ8is special.

Due to its deﬁnition, iλ8is linearly independent of the ﬁrst seven slides, and it is the only full slide

that deforms all three strands. At the same time, iλ8is the slide that resembles most the original

deﬁnition of the third Reidemeister move, which was illustrated on top of Figure 7. In contrast to

the upper nine deformations in Figure 8that seem so different from the third Reidemeister move,

iλ8conﬁrms that the slides indeed are generalizations of the third Reidemeister moves.

To deduce the matrix representation of slides given in Table 1and the multiplication Table 2

from Figure 8, it is convenient to start with the triplet iλ1,iλ2and iλ3. As mentioned, the corre-

sponding slides, which were also illustrated in Figure 7, generate an SU(2) subgroup because they

deform a pair of crossing strand segments in a way that reproduces the belt trick. The dotted circle

15

Table 1 The matrix representations for the Hermitian operators λ1...λ10 are listed. As shown in

the text, the representations follow from the moves illustrated in Figure 8. The eight SU(3)

generators are given by iλ1,...,iλ8. For these eight operators, the deﬁnition of λ8yields the

general trace relations tr λn= 0 and tr(λnλm) = 2δnm . This particular matrix representation is

called the Gell-Mann representation.

λ1=

010

100

000

, λ2=

0−i0

i0 0

000

, λ3=

100

0−1 0

000

,

λ4=

001

000

100

, λ5=

0 0 −i

000

i0 0

, λ9=

−100

000

001

,

λ6=

000

001

010

, λ7=

000

0 0 −i

0i0

, λ10 =

000

010

0 0 −1

,

and λ8=1

√3

100

010

0 0 −2

.

plays the role of the belt buckle in the belt trick. Figure 8illustrates that the same happens for the

other two triplets: each triplet is based on one crossing strand pair. In the matrix representation

of the slides given in Table 1, the undeformed strand corresponds to the column and the row con-

taining only zeros. As just explained, Figure 8implies that the squares (iλ1)2,(iλ2)2and (iλ3)2

have the diagonal values (−1,−1,0) and zero everywhere else: the belt trick acts only on the ﬁrst

strand pair. Using the commutation properties of the ﬁrst triplet, again due to the belt trick, one

thus ﬁnds that the ﬁrst two rows and columns of the ﬁrst triplet reproduce the Pauli matrices. The

matrix representation of the ﬁrst triplet is thus ﬁxed.

It is straightforward to conﬁrm from Figure 8that the concatenation of generalized slides from

the same SU(2) triplet, but around different axes, is anti-commutative, as expected from SU(2)

generators. The multiplication behaviour of the ﬁrst slide triplet is listed in the multiplication

Table 2in the nine ﬁelds on the top left. Figure 7also illustrates a difference to the usual SU(2)

multiplication table. The square of the generators iλ1,iλ2and iλ3cannot be −1, because one

strand remains undeformed, and because the crossing of the other two strands shifts against the

undeformed strand when the deformations are performed twice. Therefore, the square of each

generator is effectively equal to −1only for the two strands that are deformed, but not for the whole

structure. The diagonal values in the multiplication table follow once the matrix representation is

complete, and the deﬁnition of λ8is used.

The two additional slide triplets, also forming SU(2) subgroups, are illustrated in Figure 8.

16

Their matrix representations, given in Table 1, follow when the corresponding strand pairs are

taken into account. The matrix representations are Pauli matrices for those strand pairs that are

being deformed. The undeformed strand in each triplet yields a vanishing column and row in the

matrices. In the matrix representation, the squares of the last two triplets thus have the diagonals

(1,0,1) and (0,1,1) and have zero everywhere else, as expected. In the multiplication Table 2,

the triplets are separated by vertical lines and by thicker horizontal lines. Within each triplet, the

squares for each slide are all equal, as expected.

The matrix representations for the triplet iλ1, iλ2, iλ3, for iλ4, iλ5, iλ9, and for iλ6, iλ7, iλ10

are thus ﬁxed by Figure 8. This correspondence also ﬁxes the matrix representation for λ8, from

its deﬁnition, and for its square. In particular, all slide matrices have trace 0. As expected, the

matrices of the ﬁrst eight slides are linearly independent. The matrix representation for λ3and λ8

shows that the trace of their squares is 2 in both cases, that they commute, and above all, that they

are orthogonal to each other. In short, the trace of slide products is tr(λnλm) = 2δnm.

In short, the preceding arguments prove that Figure 8ﬁxes the speciﬁc matrix representation

of the deformations iλ1,...,iλ8that is given in Table 1. This matrix representation is called

the Gell-Mann representation. The resulting multiplication table is given in Table 2. The matrix

representation deﬁnes the SU(3) Lie algebra. Together, the matrix representation and the mul-

tiplication table also yield the matrix commutators. The commutators conﬁrm that the compact,

non-commutative, eight-dimensional Lie algebra deﬁned by the eight generators iλ1,...,iλ8, with

its three SU(2) subalgebras, is the standard SU(3) Lie algebra. The SU(3) structure constants can

be deduced from the matrix representation.

To get the SU(3) Lie group from the SU(3) Lie algebra, like for the simpler gauge groups,

the deformations of Figure 8must be generalized to arbitrary angles: one can imagine that the

crossings inside the dotted circles are deformed by an arbitrary angle around the shift-rotation

axis. Such deformations are called generalized slides in the following.

Generalized slides obey the group axioms and form a group: they can be concatenated (mul-

tiplied), the concatenation is associative, there is a neutral element (no rotation at all), and each

generalized slide has an inverse (the inverse rotation of the segment pair). More precisely, gen-

eralized slides form a Lie group: they form a compact manifold, because their parameters are

(real) angles and their concatenations behave nicely on this manifold. The full set of generalized

slides is parametrized by eight angles or real numbers. The Lie group of generalized slides is thus

eight-dimensional. Together with the SU(3) Lie algebra, all these properties imply that generalized

slides deﬁned with Figure 8form the Lie group SU(3).

Also for slides, the description with deformations, the unitarity property of the group elements,

and the Hermitian property of the Gell-Mann matrices all imply each other. Modelling gauge

interactions with deformations thus explains the ‘U’ of SU(3). Without the tethers, unitarity would

not arise. The impenetrability of strands implies vanishing traces of the representing matrices,

17

Table 2 As shown in the text, Figure 8implies the following multiplication table, using the

concatenation of deformations as multiplication. Barring the rows and columns for λ9and λ10 ,

and multiplying each slide λnby i, yields the multiplication table of the generators of SU(3). The

linearly dependent slides λ9=−λ3/2−λ8√3/2and λ10 =−λ3/2 + λ8√3/2do not yield

generators. These two slides are used to construct λ8using λ8= (λ10 −λ9)/√3. The three

SU(2) subgroups are generated by the triplet λ1,λ2, and λ3, the triplet λ4,λ5and λ9, and the

triplet λ6,λ7, and λ10. Despite the ﬁrst impression, λ2

4=λ2

5=λ2

9and λ2

6=λ2

7=λ2

10.

λ1λ2λ3λ4λ5λ9λ6λ7λ10 λ8

λ12/3iλ3−iλ2λ6/2−iλ6/2−λ1/2λ4/2−iλ4/2λ1/2λ1/√3

+λ8/√3 +iλ7/2 +λ7/2 +iλ2/2 +iλ5/2 +λ5/2 +iλ2/2

λ2−iλ32/3iλ1iλ6/2λ6/2−iλ1/2−iλ4/2−λ4/2−iλ1/2λ2/√3

+λ8/√3−λ7/2 +iλ7/2−λ2/2 +λ5/2−iλ5/2 +λ2/2

λ3iλ2−iλ12/3λ4/2−iλ4/2−1/3−λ3/3−λ6/2iλ6/2−1/3 + λ3/3λ3/√3

+λ8/√3 +iλ5/2 +λ5/2 +λ9/3−iλ7/2−λ7/2 +λ10/3

λ4λ6/2−iλ6/2λ4/2 2/3 + λ3/2−iλ9iλ5λ1/2iλ1/2−λ4/2−λ4/2√3

−iλ7/2−λ7/2−iλ5/2−λ8/2√3 +iλ2/2−λ2/2−iλ5/2−i√3λ5/2

λ5iλ6/2λ6/2iλ4/2iλ92/3 + λ3/2−iλ4−iλ1/2λ1/2iλ4/2i√3λ4/2

+λ7/2−iλ7/2 +λ5/2−λ8/2√3 +λ2/2 +iλ2/2−λ5/2−λ5/2√3

λ9−λ1/2iλ1/2−1/3−λ3/3−iλ5iλ42/3 + 2λ3/3λ6/2iλ6/2−1/3−λ9/3−1

−iλ2/2−λ2/2 +λ9/3 +λ9/3−iλ7/2 +λ7/2 +λ10/3 +λ10

λ6+λ4/2iλ4/2−λ6/2λ1/2iλ1/2λ6/2 2/3−λ3/2iλ10 −iλ7−λ6/2√3

−iλ5/2 +λ5/2 +iλ7/2−iλ2/2 +λ2/2 +iλ7/2−λ8/2√3−i√3λ7/2

λ7iλ4/2−λ4/2−iλ6/2−iλ1/2λ1/2−iλ6/2−iλ10 2/3−λ3/2iλ6i√3λ6/2

+λ5/2 +iλ5/2−λ7/2−λ2/2−iλ2/2 +λ7/2−λ8/2√3−λ7/2√3

λ10 −λ1/2−iλ1/2−1/3 + λ3/3−λ4/2−iλ4/2−1/3−λ9/3iλ7−iλ62/3−λ3/3 1

+iλ2/2−λ2/2−λ10/3 +iλ5/2−λ5/2 +λ10 /3 +λ9/3 +λ9

λ8λ1/√3λ2/√3λ3/√3−λ4/2√3−i√3λ4/2−1−λ6/2√3−i√3λ6/2 1 2/3

+i√3λ5/2−λ5/2√3 +λ10 +i√3λ7/2−λ7/2√3 +λ9−λ8/√3

implies the determinant +1, and thus explains the ‘S’ of SU(3). The three strands involved in the

slides explain the ‘(3)’ of SU(3).

18

6 Checking the derivation of SU(3)

There are several ways to check that the deduction of SU(3) from Figure 8is correct. First of all,

Figure 8contains three SU(2) subalgebras, due to the three belt buckles that are contained in it.

The three SU(2) subalgebras are rotated by the angle ±2π/3with respect to each other, around

the axis deﬁned by the direction of observation. Strands thus illustrate the threefold C3symmetry

of SU(3), as expected. (Due to the squashing of the graphs in Figure 8, the symmetry is not fully

obvious. It is more obvious in Figure 7.) The four slides on the rightmost column of Figure 8are

all part of the centre of SU(3). In particular, the linear dependent slides λ3,λ9, and λ10 illustrate

the C3symmetry of the centre of SU(3) – again as expected.

Secondly, in each triplet, the squared slides leave one strand undeformed and shift the crossing

of the other two strands towards the undeformed strand. Indeed, in the multiplication table for

the ﬁrst triplet, the square of each slide involves λ8. Without the shift, λ8would not arise in the

table. Likewise, the squares in the other two triplets involve λ3and λ8. This is as expected from

the threefold symmetry of SU(3). The multiplication behaviour is the same as for the ﬁrst triplet,

with the diagonal products transformed by a rotation by ±2π/3. Again, the result is as expected.

Thirdly, the product value of a slide with itself can be checked using Figure 8. This requires

the deﬁnition of scalar multiplication and addition. This step was not necessary in the case of pure

SU(2) and is the reason that the strand realization of SU(3) was overlooked for a long time.

In the strand tangle model, the addition of strand-pair deformations is realized when the two

deformations are applied at the same time. The scalar multiplication of a strand deformation is

realized by multiplying the corresponding rotation angle of the circled region (the belt buckle).

(The mentioned equivalence between 2πand −1, and between πand iare used.) As an example,

the deﬁnition λ8= (λ10−λ9)/√3yields the deformation illustrated at the very bottom of Figure 8:

above all, it switches the orientation of the central triangle.

Using the deﬁnitions of addition and scalar multiplication, the products of each slide with itself

can be checked. For the products on the diagonal of the ﬁrst triplet – given in Figure 7and in

Figure 8– the matrix multiplication yields λ2

1=λ2

2=λ2

3= 2/3 + λ8/√3 = 2/3−λ9/3 + λ10/3.

The numbers are not deduced in a straightforward way directly from the deformations in Figure 8;

but the deformations do show that the squares of the slides in the ﬁrst triplet are independent of λ3.

Therefore the squares of the ﬁrst triplet are a linear combination of the identity and λ8. (This linear

combination narrows the central triangle between the three strands along the west-east direction

and leaves the opposite strand untouched.) The numbers in the deﬁnition of iλ8– equivalently,

the numbers in its diagonal matrix representation – explain the three entries on the diagonal of the

multiplication table for the ﬁrst slide triplet. The values of the diagonal of the multiplication table

for the other two triplets follow after rotation by ±2π/3around the direction of observation. The

square of λ8follows. In other words, the squares of all slides are ﬁxed by Figure 8.

19

Finally, Figure 8allows several additional checks of the slide multiplication table. Figure 8

implies that the four slides iλ3,iλ9,iλ10 and iλ8in the last column all commute among each other.

This is reproduced in the multiplication table.

Compared to SU(2), which is anti-commutative, SU(3) is more strongly non-commutative.

This is best seen in the products between the ﬁrst slides from different triplets. An example is the

difference between the products λ1λ4and λ4λ1. Exploring the concatenation of the corresponding

slides shows that the two products are not the negative of each other. This happens because each

belt trick operation also shifts the belt (the region inside the dotted circle), and the shifts destroy

the anti-commutation for the cases that the two slides are from different triplets. Due to these

shifts, the product λ1λ4yields a linear combination of the slides of the remaining triplet; and the

product differs from the product λ4λ1. This is as expected. Slides thus do not commute in general.

And like pokes, slides generate a Yang-Mills theory [10].

As a remark, the group SU(3) can also be deduced from deformations of a single strand seg-

ment, instead of deformations of crossing strand pairs. However, the images are less pedagogical.

Possibly, an even more pedagogical set of eight deformations yielding SU(3) can be found.

In contrast to strands, number ﬁelds do not explain the gauge groups: even though U(1) are the

unit complex numbers and SU(2) are the unit quaternions, the unit octonions do not form a group

and have no simple relation to SU(3).

In short, the third Reidemeister move, the slide, naturally yields eight deformations that gener-

ate the Lie group SU(3) of generalized slides. In the strand description of wave functions, particles

and interactions, slides play an important role. A slide changes the phase of a fermion and thus

models an interaction. Slides and their Lie group SU(3) can be used to deﬁne a model for the strong

nuclear interaction, for the gluons, and for the colour charge, as explained elsewhere [10]. The

quark model, Regge trajectories, glueballs, the lack of CP violation in the strong interaction, and

the strong coupling constant arise naturally. Strands fully reproduce quantum chromodynamics

and predict that no measurable deviation from quantum chromodynamics will ever be observed.

7 The possible gauge groups in nature

Explaining the gauge groups U(1), SU(2) and SU(3) as the result of strand deformations is at-

tractive for several reasons. First, in the research literature, no other, ab initio explanation of the

gauge groups that agrees with all experiments – and in particular, that does not add additional,

unobserved ﬁelds – has been published. So far, the strand explanation is unique and unmodiﬁable.

Secondly, the gauge groups arise as consequences of the same idea with which Dirac explained

spin 1/2and fermion behaviour [5], with which Battey-Pratt and Racey explained the Dirac equa-

tion [6,9], and with which general relativity can be deduced [11,12]. The strand explanation is

simple,consistent and complete.

20

Thirdly, the explanation of the gauge groups predicts the lack of additional gauge groups in

nature, and in particular the lack of larger, uniﬁed gauge groups. For example, strands imply

that gauge groups like SU(5), SO(10) or E8 do not exist in nature, and neither does any other

Yang-Mills theory. Again, the explanation with strands agrees with all experiments performed so

far [13]. The strand tangle model is correct.

Fourthly, in a similar way that the classiﬁcation of tangle deformations leads to the gauge

interactions, also the classiﬁcation of rational tangle structures leads to the observed elementary

particles, and to no additional ones [7,8]. Dark matter is predicted not to be made of unknown

elementary particles. The strand tangle model is predictive.

Finally, strands imply that the fundamental constants – coupling constants, mixing angles and

particle mass ratios – have unique and calculable values. In particular, strands imply that the

statistics of their shape ﬂuctuations allow calculating these values. The ﬁrst rough estimates agree

with data [9,10]. The strand tangle model is testable.

In short, the strand tangle model, in contrast to other approaches, implies the lack of additional

gauge groups in nature. In particular, this implies the lack of a uniﬁed gauge group. Strands also

imply the lack of any other new physics. Strands further imply unique values for the fundamental

constants that are of the order of the measured values. Due to the wide-ranging implications

of the strand tangle model, a thorough check – both mathematical and experimental – of all its

consequences should be performed.

8 Conclusion

The three Reidemeister moves – twists, pokes and slides – have been shown to generate the Lie

groups U(1), SU(2), and SU(3), once the moves are interpreted as deformations of strand tangles

that model particles, wave functions and interactions. Because Reidemeister proved that every

tangle deformation is a combination of the three moves only, the strand tangle model implies the

lack of any other gauge group in nature. So far, this conclusion agrees with all observations. It ap-

pears that the explanation of gauge theory using strands and their deformations is unique, correct,

simple, ab initio, consistent, complete, unmodiﬁable, predictive and testable. The wide explana-

tory power of the strand model suggests exploring it as an approach to uniﬁcation. Experimental

observation of any new physics beyond the standard model with massive Dirac neutrinos with

PMNS mixing would falsify the model. Comparing high-precision calculations of the coupling

constants and the other fundamental constants to the measured values will provide deﬁnite tests of

the model.

21

9 Acknowledgements and declarations

The author thanks Thomas Racey, Michel Talagrand, Jason Hise, John Baez, Sebastian Meyer and

Isabella Borgogelli for stimulating discussions and support. This work was supported partly by a

grant from the Klaus Tschira Foundation. The author declares that he has no conﬂict of interest

and no competing interests. No additional data are associated with the text.

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