From tangles of strands to elementary particles, wave functions, gauge groups and the standard model, as well as to space, curvature and general relativity

Abstract

The slides present the only known mathematical explanation of the gauge groups that agrees with observations. The explanation also yields the only known model for emergent wave functions and the only known explanation of the elementary particle spectrum.

Several animations are included. Click and download the pdf to see them.

The slides are based on the publication https://www.researchgate.net/publication/374297019
and on the longer pedagogical preprint https://www.researchgate.net/publication/361866270

Full text

Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 1 / 39
Nature is topological: it plays cat’s cradle
From tangles of strands to elementary particles,
wave functions, gauge groups and the standard model,
as well as to space, curvature and general relativity
Christoph Schiller, April 2024

The State of Physics: 9 Lines Describe Nature
Quantum of Action
OPhysics in 9 lines
ODirac’s trick
OSpin 1/2
OSpin animation
OFermions
OFermion animation
ODirac’s letter
OFundamental
principle
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 2 / 39
(1) dW=0 Action W=Ldtis minimized in local motion. The other lines fix the two funda-
mental Lagrangians L.
(2) vcLocal energy speed vis limited by the speed of light c. This implies special relativity
and restricts the possible Lagrangians.
(3) Fc4/4GLocal force Fis limited by cand by the gravitational constant G. This implies general
relativity and fixes its Lagrangian.
(4) W

Action Wis never smaller than the quantum of action . This implies quantum theory
and restricts the possible Lagrangians.
(5) Skln
ln
ln2Entropy Sis never smaller than ln 2 times the Boltzmann constant k. This implies
thermodynamics.
(6) U(1) is the gauge group of the electromagnetic interaction. It yields its Lagrangian.
(7) SU(3) and
broken SU(2)
are the gauge groups of the two nuclear interactions, yielding their Lagrangians.
(8) 18 elementary
particles
– gauge bosons, the Higgs boson, quarks, leptons, and the undetected graviton – with
all their quantum numbers, make up everything in nature and, with their interactions,
fix the standard model Lagrangian.
(9) Finally, 27 numbers – dimensions, cosmological constant, coupling constants, particle mass ratios, mix-
ings and phases – complete the two fundamental Lagrangians. They determine all
observations and all colours.
(Link to details and to a paper that summarizes about half a million publications in the past 50 years.)
Lines 6, 7, 8 and 9 need explanations. This talk explains lines 6, 7 and 8.

Dirac’s Lecture Trick – According to Penrose
Quantum of Action
OPhysics in 9 lines
ODirac’s trick
OSpin 1/2
OSpin animation
OFermions
OFermion animation
ODirac’s letter
OFundamental
principle
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 3 / 39
R. Penrose (Dirac’s student) & W. Rindler, Spinors and space-time, vol. I (1984).
The scissors represent a spin 1/2 fermion.
The chair represents the cosmological horizon.
Only a (scissor) rotation by 4πleads back to the original situation. 2πdoes not.
Is every particle tethered (attached) to the cosmological horizon? Yes.

Strands and Belts Explain Spin 1/2
Quantum of Action
OPhysics in 9 lines
ODirac’s trick
OSpin 1/2
OSpin animation
OFermions
OFermion animation
ODirac’s letter
OFundamental
principle
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 4 / 39
particle,
i.e.,
tangle
core
Dirac's belt trick or string trick: Double tethered particle rotation is no rotation.
Resulting observation. Time averaging unobservable tethers but observable
crossing switches leads to a probability density and phase:
phase
move
all
tethers
move
lower
tethers
move
upper
tethers
move
tethers
sideways
like
start
rotate
particle
twice in
any
direction
tethers tethers
probability
density
phase phase
Core/particle rotation by 4πis equivalent to no rotation, for 2 or more strands.

Dirac’s Trick Implies That Spin Is Rotation
Quantum of Action
OPhysics in 9 lines
ODirac’s trick
OSpin 1/2
OSpin animation
OFermions
OFermion animation
ODirac’s letter
OFundamental
principle
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 5 / 39
© Jason Hise
The cubic centre represents a lepton tangle core.
As illustrated below, leptons have six tethers.
Dirac’s trick works with any number of tethers equal or larger than 3.
A spinning particle is a rotating tangle core.

Strands and Belts Explain Fermions
Quantum of Action
OPhysics in 9 lines
ODirac’s trick
OSpin 1/2
OSpin animation
OFermions
OFermion animation
ODirac’s letter
OFundamental
principle
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 6 / 39
move
tethers
only
particles,
i.e., tangle
cores
move
tethers
only
The fermion trick: Double tethered particle exchange is no exchange.
exchange
particle
cores
twice
like
start
The trick also works if some or all the strands connect one tangle core to the other core.
tethers
tethers
Double exchange is no exchange.
The Dirac trick yields the fermion trick.

Fermion Behaviour Allows Orbiting Particles
Quantum of Action
OPhysics in 9 lines
ODirac’s trick
OSpin 1/2
OSpin animation
OFermions
OFermion animation
ODirac’s letter
OFundamental
principle
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 7 / 39
© Antonio Martos
The fermion trick works for any number of tethers.
Spin 1/2 particles are fermions. This is (half) the spin-statistics theorem.

Paul Dirac’s Letter to Martin Gardner
Quantum of Action
OPhysics in 9 lines
ODirac’s trick
OSpin 1/2
OSpin animation
OFermions
OFermion animation
ODirac’s letter
OFundamental
principle
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 8 / 39
M. Gardner, Riddles of the Sphinx and Other Mathematical Puzzle Tales (1987), page 47.
Rotations of tethered particles produce crossing switches.
The essence of Dirac’s trick are crossing switches.
Acrossing switch is a change of overpass and underpass:
Therefore, crossing switches yield Planck’s quantum of action .
(L. Kauffman, 1987)

Crossing Switches Define Planck’s 
Quantum of Action
OPhysics in 9 lines
ODirac’s trick
OSpin 1/2
OSpin animation
OFermions
OFermion animation
ODirac’s letter
OFundamental
principle
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 9 / 39
Strand description: Resulting
observation:
A fundamental
event localized
in space-time
within Planck scales
The fundamental, Planck-scale principle of the strand tangle model
W=
Δt≥4G/c5
Δl≥4G/c3
S=kln 2
tt+Δt
Strands have Planck radius. Strands are unobservable, impenetrable and
featureless: no mass, no tension, no torsion, no branches, no fixed length, no ends.
Acrossings is the region of the smallest distance between two strands.
Every event is a crossing switch characterized by .
All observables are defined and measured in terms of crossing switches.
Thesis: This fundamental principle implies all of physics.
The principle implies general relativity (via Fc4/4G) and the standard model,
with the three gauge groups and the known particles. And not more.
(Link to details and publications.)

Rational 3d Tangles Are Special
Quantum of Action
Wave Functions
ORational 3d tangles
OElementary
fermions
OWave functions
OSpinning electron
Gauge interactions
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 10 / 39
Knotted tangles
Knots
trivial non-trivial
trivial non-trivial
core
tethers
an open knot
a prime tangle
trefoil
knot
figure-8
knot
above
tether
above
paper plane
above
below
below
tether
below
paper plane
Rational
(
3d
)
tangles – or
(
3d
)
braids
Links
trivial non-trivial
trivial
tethers
core
non-trivial
Hopf
link
Borromean
rings
Only rational 3d tangles reproduce particle reactions and transformations.
In the strand model, particles are rational 3d tangles.

Elementary Fermions Are Rational 3d Tangles
Quantum of Action
Wave Functions
ORational 3d tangles
OElementary
fermions
OWave functions
OSpinning electron
Gauge interactions
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 11 / 39
Leptons - `cubic' tangles made of three strands along cordinate axes (only simplest family members)
Parity P = +1, Baryon number B = +1/3, Spin S = 1/2
Charge Q = –1/3
Quarks - `tetrahedral' tangles made of two strands with four tethers (only simplest family members)
Charge Q = +2/3
d quark
above plane
tether in
paper plane
below paper
plane
below paper
plane
u quark
above
below
below
tether in
paper plane
s quark
above
below
below
c quark
above
below
below
t quark
above
below
below
b quark
above
below
below
tether in
paper plane
electron
Q = –1, S = 1/2
above
above
paper
plane
above
below
below
below paper plane
electron neutrino
Q = 0, S = 1/2 below
paper
plane
below
below
above paper plane
above
above
muon
Q = –1, S = 1/2
above
above
paper
plane
above
below
below
below paper plane
muon neutrino
Q = 0, S = 1/2 below
paper
plane
below
below
above
above
above paper plane
tau
Q = –1, S = 1/2
above
above
paper
plane
above
below
below
below paper plane
tau neutrino
Q = 0, S = 1/2
below
paper
plane
below
below
above
above
above paper plane
‘Elementary’ means
1 to 3 strands.
‘Fermion’ means
localizable tangle
with 2 or more
strands.
These simplest
tangles reproduce
all quantum numbers.
No additional
elementary fermions
are possible.
No other explanation
of the particle
spectrum exists.
(Pedagogical link.)

Wave Functions Are Crossing Densities
Quantum of Action
Wave Functions
ORational 3d tangles
OElementary
fermions
OWave functions
OSpinning electron
Gauge interactions
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 12 / 39
The strand tangle model for wave functions
tangle
core
crossing
midpoint
spin
axis
Crossing midpoints
with their amplitudes
and local phases
Step 1: Take the
crossing midpoints
and their phases
of the above tangle.
Step 2: Take the time
average of all crossing
midpoints and phases to
get the wave function.
Wave function
amplitude
and central
phase
central
phase
The black dot specifies the crossing position,
the shortest distance s determines the
crossing amplitude, and the angle 
defines the crossing phase.
s amplitude

s amplitude
position
Spinning
electron tangle
Spin S = 1/2
tether
spin
axis
flagflag
flag
crossing axis
phase orientation
around crossing axis
, 

Tangles are skeletons
of wave functions.
Blurred tangles follow
the free Dirac equation.
(Battey-Pratt & Racey, 1980.)
Crossings have amplitudes (inverse distance) and phases.
Crossing densities of fluctuating tangles are wave functions: they yield Hilbert
spaces, interference, decoherence, collapse, and entanglement. (Pedagogical link.)
Dirac’s equation is the infinitesimal version of Dirac’s trick.

The Spinning Electron (slightly incorrect)
Quantum of Action
Wave Functions
ORational 3d tangles
OElementary
fermions
OWave functions
OSpinning electron
Gauge interactions
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 13 / 39
Rotation details depend
on the speed:
mass arises.
Mass calculation requires
estimating the number
of Dirac tricks per time.
A challenge! (Link with prizes.)
In any case,
the masses of elementary
particles are small:
mmPl.

Interactions Are Tangle Core Deformations
Quantum of Action
Wave Functions
Gauge interactions
OInteractions
OReidemeister
moves 1
OReidemeister
moves 2
OU(1)
OSU(2)
OSU(3)
OGell-Mann matrices
OElementary
bosons
OSM Lagrangian 1
OSM Lagrangian 2
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 14 / 39
time average
of crossing
switches
above
above
paper
plane
above
below
below
below paper
plane
electron
Q = +1
probability
density
spin
Observation A core deformation
phase
Tangle model
phasephase
spin spin
Free propagating particles are cores that rotate:
Core rotation axis →spin axis
Core orientation →phase of wave function
Tether deformations for rigid cores →space-time symmetries
Interacting fermions are cores being deformed:
Core deformations change the phase →interactions
Freedom in the definition of phase →freedom of gauge
Surprise: All observable deformations can be built from 3 basic types.

Animated Reidemeister Moves
Quantum of Action
Wave Functions
Gauge interactions
OInteractions
OReidemeister
moves 1
OReidemeister
moves 2
OU(1)
OSU(2)
OSU(3)
OGell-Mann matrices
OElementary
bosons
OSM Lagrangian 1
OSM Lagrangian 2
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 15 / 39
Double click.
Reidemeister moves are related to crossing switches.
Reidemeister moves in tangles cores are thus physically observable.
Reidemeister moves are the only physically observable deformations.

Reidemeister Moves Classify Interactions
Quantum of Action
Wave Functions
Gauge interactions
OInteractions
OReidemeister
moves 1
OReidemeister
moves 2
OU(1)
OSU(2)
OSU(3)
OGell-Mann matrices
OElementary
bosons
OSM Lagrangian 1
OSM Lagrangian 2
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 16 / 39
Every tangle core deformation is built from three basic types:(Reidemeister 1926)
Reidemeister move I
or twist
Reidemeister move II
or poke
Reidemeister move III
or slide
Twists generate U(1), pokes generate SU(2), parity violation and symmetry
breaking, while slides generate SU(3).(Schiller 2009, 2019, 2024 link.)
Gauge interactions are (statistical) crossing transfers:
virtual vacuum
photon
Weak interaction is
poke transfer:
Strong interaction is
slide transfer:
Electromagnetic interaction is
twist transfer:
virtual
gluon
fermion fermion
vacuum
virtual
(unbroken)
weak boson
vacuum
fermion

Twists Generate Local U(1)
Quantum of Action
Wave Functions
Gauge interactions
OInteractions
OReidemeister
moves 1
OReidemeister
moves 2
OU(1)
OSU(2)
OSU(3)
OGell-Mann matrices
OElementary
bosons
OSM Lagrangian 1
OSM Lagrangian 2
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 17 / 39
A double twist of the encircled segment can be rearranged to an untwisted strand,
keeping the encircled segment fixed in space: no twist:
In a fermion, the twist around a given axis thus generates a local U(1) Lie group.
The twist, or first Reidemeister move, is related to a crossing switch:
Twists, performed by rotating the encircled segment, are thus observable.
ππ
axis
π
first
twist
second
twist
rearrangement no
twist
Twists rotate the dotted circle by π.Generalized twists rotate
thedottedcirclebyarbitrary angles. They form the local Lie group U(1).
Rotating twists also yield a model for the photon. (More later on.)

Pokes Generate SU(2) via the Belt Trick
Quantum of Action
Wave Functions
Gauge interactions
OInteractions
OReidemeister
moves 1
OReidemeister
moves 2
OU(1)
OSU(2)
OSU(3)
OGell-Mann matrices
OElementary
bosons
OSM Lagrangian 1
OSM Lagrangian 2
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 18 / 39
The poke, or second Reidemeister move, on pairs of strands generates an SU(2)
Lie group, because the three rotations by generate the algebra of SU(2):
π
πx
y
π
z
Pokes, like belts, yield the Pauli matrices, i.e., the Lie algebra of SU(2):
τx=iσx=i01
10
,τ
y=iσy=i0−i
i0,τ
z=iσz=i10
0−1
Generalized pokes, by arbitrary angles, yield the full local Lie group SU(2).
Maximal parity violation and SU(2) breaking also follow (see bonus material).

Slides Generate Three Belt Tricks and SU(3)
Quantum of Action
Wave Functions
Gauge interactions
OInteractions
OReidemeister
moves 1
OReidemeister
moves 2
OU(1)
OSU(2)
OSU(3)
OGell-Mann matrices
OElementary
bosons
OSM Lagrangian 1
OSM Lagrangian 2
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 19 / 39
Slides, orthird Reidemeister moves, acting on strand pairs in three-strand structures, canbe
generalizedtothegeneratorsoftheLiegroupSU(3).
iλ1
iλ4
iλ6
π
π
π
Slides rotate the
dottedcirclebyπ.
The deformations
allow reading off
the matrix
representations
(see next page).
λ3,λ9and λ10
are not linearly
independent.
Traditionally,
λ3and λ8are used.
λ8is the slide
prototype.

Slides Generate SU(3)’s Gell-Mann Matrices
Quantum of Action
Wave Functions
Gauge interactions
OInteractions
OReidemeister
moves 1
OReidemeister
moves 2
OU(1)
OSU(2)
OSU(3)
OGell-Mann matrices
OElementary
bosons
OSM Lagrangian 1
OSM Lagrangian 2
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 20 / 39
λ1=⎛
⎝
010
100
000
⎞
⎠,λ
2=⎛
⎝
0−i0
i00
000
⎞
⎠,λ
3=⎛
⎝
100
0−10
000
⎞
⎠,
λ4=⎛
⎝
001
000
100
⎞
⎠,λ
5=⎛
⎝
00−i
00 0
i00
⎞
⎠,λ
9=⎛
⎝−100
000
001
⎞
⎠,
λ6=⎛
⎝
000
001
010
⎞
⎠,λ
7=⎛
⎝
00 0
00−i
0i0
⎞
⎠,λ
10 =⎛
⎝
00 0
01 0
00−1
⎞
⎠,
and λ8=1
√3
⎛
⎝
10 0
01 0
00−2
⎞
⎠.
Of the ten slide deformations, only the first eight are linearly independent.
These eight deformations yield the Gell-Mann matrices.
The eight deformations generate the algebra of SU(3) – and describe gluons.
These eight generators also yield the relations tr λn=0and tr(λnλm)=2δnm.
SU(3) has three linear independent SU(2) subgroups – one in each row.
Generalized slides, by arbitrary angles, yield the full Lie group SU(3). (Publication link.)

Elementary Bosons Follow
Quantum of Action
Wave Functions
Gauge interactions
OInteractions
OReidemeister
moves 1
OReidemeister
moves 2
OU(1)
OSU(2)
OSU(3)
OGell-Mann matrices
OElementary
bosons
OSM Lagrangian 1
OSM Lagrangian 2
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 21 / 39
3 strands: eight gluons
2 strands: W3 (before SU(2) symmetry breaking)
Z boson
3 strands: Higgs boson
wavelength
wavelength
1 strand: photon
W boson
Virtual bosons:
Weak (real) vector bosons after SU(2)
symmetry breaking, thus massive
(only the simplest family members)
2 strands: W1, W2 (before SU(2) symmetry breaking)
wavelength
wavelength
Elementary bosons are simple configurations of 1, 2 or 3 strands that propagate:
Spin
S = 1
S = 1
S = 1
S = 0
S = 1
S = 1
S = 1
2 strands: graviton
wavelength
S = 2
Real bosons:
‘Elementary’ means
1, 2 or 3 strands.
‘Boson’ means
unlocalizable
tangle.
The gauge bosons
tangles reproduce
all quantum numbers.
No additional
gauge bosons
are possible.
No other explanation
of the gauge
spectrum exists.
(Pedagogical link.)

Vertices 1
Quantum of Action
Wave Functions
Gauge interactions
OInteractions
OReidemeister
moves 1
OReidemeister
moves 2
OU(1)
OSU(2)
OSU(3)
OGell-Mann matrices
OElementary
bosons
OSM Lagrangian 1
OSM Lagrangian 2
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 22 / 39
time average
of crossing
switches
Observed
Feynman diagram
Tangle model
time
c quark W
s quark
s quark s quark Z
t2
time
t1
s quark s quark
s quark
W
c quark
Z
time average
of crossing
switches
Observed
Feynman diagram
time
muon neutrino W
muon
Z
muon muon Z
time
muon muon
muon
W
muon neutrino
time W W
time
Z
photon photon
W W
time
W W
W W
time
photon photon
W W
WW
time
Z photon
W W
Z photon
time
W W
ZZ Z Z
Discs indicate a pair of tethers
or a long tether
Tangle model
photon photon
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
WW
WW
WW
WW
Z
WW
WW
The rational 3d
particle tangles
limit the possible
interaction vertices.
Due to the tangle
topology, only
triple or quadruple
vertices arise, but
no fourfold fermion
vertices.
Renormalizability
is thus automatic
in the tangle model.

Vertices 2
Quantum of Action
Wave Functions
Gauge interactions
OInteractions
OReidemeister
moves 1
OReidemeister
moves 2
OU(1)
OSU(2)
OSU(3)
OGell-Mann matrices
OElementary
bosons
OSM Lagrangian 1
OSM Lagrangian 2
Gravitation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 23 / 39
time
Observed
Feynman diagram
Higgs
Z
Z Z
Z Z
Higgs Higgs
ZZ time
Higgs Higgs
time
Higgs Higgs
time
Higgs Higgs
Higgs
W W
time Higgs
time
Higgs Higgs
Higgs
Higgs Higgs
Higgs Higgs
Higgs Higgs
time
time
quark gluon
quark
time
quark gluon
quark
time
Observed
Feynman diagram
Tangle model
Higgs
fermion
fermion
s quark
s quark Higgs
time average
of crossing
switches
Tangle model
vacuum
gluon gluon
gluon gluon
gluon gluon
Higgs
W W
gluon
gluon
gluon gluon
gluon gluon
gluon gluon
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
the quark core is rotated
by 2/3 around the
horizontal axis.
Higgs
Higgs Higgs
Higgs Higgs
time average
of crossing
switches
t2
t1
t2
t1
t2
t1
t2
t1
t2
t1
Discs indicate a pair of tethers
or a long tether
Z
WW
WW
The rational 3d
particle tangles
also yield Higgs
self-interactions.
No vertex of
the standard
model
is missing.
Due to the tangle
topologies, no
additional
vertices arise.
The full standard
model Lagrangian
arises.

A Planck-Scale Model of Almost Everything
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
OEverything strands
OGravitation
OBlack hole rotation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 24 / 39
Observation:
Nothing
(for long
observation
times)
Virtual pairs
(for short
observation
times)
The flat vacuum – a homogeneous strand aggregate
time average
of crossing
switches
Observation:
Curved space
with non-
trivial metric
Curved vacuum – an irregular strand aggregate
A black hole horizon – a weave of strands
Observation after time average of crossing switches:
a horizon, i.e., a thin spherical cloud, with mass,
moment of inertia, entropy, and temperature.
time average
of crossing
switches
A particle – a tangle of strands
time average
of crossing
switches
Observation:
Probability
density
Black dots show
the midpoints
of crossings.

A Planck-Scale Model for Black Holes
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
OEverything strands
OGravitation
OBlack hole rotation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 25 / 39
smallest
area
neighbouring
strand
with
additional
crossing
Black hole horizon in the strand conjecture:
side view / cross section view from above
Observed h
a thin spherical cloud ring
additional
crossing
The effective number nof possible microstates per smallest area:
n=2+1
2! +1
3! +1
4! +1
5! +... =e=2.71828...
yields an entropy value Sthat depends on the area A:(Schiller 2009, 2019, 2023)
S
k=A
4G/c3−O(ln A
4G/c3)
The fundamental principle implies black hole entropy, energy, temperature –
and evaporation: strands detach. Strands imply pure general relativity.
Strands imply force Fc4/4G,power Pc5/4G,mass/length
m/lc2/4G, etc. Strands again imply pure general relativity.
Thus, no singularities, negative energy regions, wormholes, black hole hair,
torsion, time-like loops, running of G, or new quantum gravity effects.

Black Holes Can Rotate
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
OEverything strands
OGravitation
OBlack hole rotation
Conclusion
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 26 / 39
©JasonHise.
Strands are not observable, only crossing switches are.
Black holes have a finite moment of inertia; mass is distributed over the horizon.

The Main Results
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
OResults
OExp. predictions
OMath Challenges
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 27 / 39
Only fluctuating tangles of strands explain wave functions.
Only fluctuating tangles of strands explain elementary particles – and their
quantum numbers and properties – from tangle classification.
Only fluctuating tangles of strands explain the gauge groups – and all the
interaction properties – using the Reidemeister moves.
The fascinating aspect is due to the simplicity of the fundamental principle and
to the uniqueness of the explanations:
•The fundamental principle implies only observed particle physics.
•The fundamental principle implies only observed general relativity.
•Only the fundamental principle provides these explanations.
•There is no way to modify or to generalize the fundamental principle or the
tangle model – and their predictions.

Predictions – Beyond The Standard Model
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
OResults
OExp. predictions
OMath Challenges
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 28 / 39
GPlanck length and Planck time are the smallest measurable intervals.
Space is neither continuous nor discrete. No new quantum gravity effects.
G3 dimensions. No supersymmetry. No non-commutative space.
GPlanck momentum and energy are the highest measurable values for
elementary particles. c4/4Gand c5/4Gare maximum force and luminosity.
Maximum values for probability densities, electric fields, magnetic fields,
strong and weak fields exist. No trans-Planckian effects.
G3 gauge interactions. Only. They are fundamental. No GUT.
G3 generations. No new particles. No unknown elementary dark matter.
No axions, no WIMPS, no sterile neutrinos, no monopoles, etc.
GNo measurable deviations from the standard model. Only known
Feynman diagrams. Scattering amplitudes, running, g−2, and electric
dipole moments are as predicted. No proton decay. No baryon number
violation. CPT holds. Dirac neutrinos with normal mass order.
GNo physics beyond the standard model with massive Dirac neutrinos.
And
!GMasses, mixing angles and coupling constants can be calculated.

Mathematical Outlook And Challenges
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
OResults
OExp. predictions
OMath Challenges
Bonus Material
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 29 / 39
Prove, clarify or disprove:
•No visualization of SU(2) or SU(3) without tethers is possible in 3 dimensions.
•No visualization of SU(n) with strands (or without strands) for n > 3 is possible
in 3 dimensions. (This has profound consequences for physics.)
•The rational 3d tangle classification is mathematically complete and leaves no
room for additional elementary fermions or bosons.
•The rational 3d tangle classification is mathematically complete and leaves no
room for additional defects in space that are neither fermions nor bosons.
Determine:
•How does the probability of belt-trick-like rotation for a tethered ball depend
on the chirality and size of the tethered structure and on the number of ropes?
Use ideas from hydrodynamics of viscous liquids.
•Use the result to estimate neutrino masses. Ideally, before they are measured.
•Calculate the three gauge coupling constants from the average tangle shape.
Earn prizes for specific math problems from geometric knot theory:
•See www.motionmountain.net/charge-mass.html

The Universe
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
OThe universe
OPath integrals
ODirac’s equation
OQuark generations
OLepton generations
OTight Eletron
OElectrons and
positrons
OMore diagrams
OReferences 1
OReferences 2
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 30 / 39
Nature is a wobbly criss-crossing strand woven into the night sky.
The universe plays cat’s cradle.
The strand conjecture:
time
cosmological
horizon
The early expanding universe
Observations:
time average
of crossing
switches
none
Cosmological
horizon (dark)
Physical space or
cosmic vacuum (white);
made of densely packed
untangled strands;
observable, locally
curved, flat on average
Background
space (grey):
not physical,
not observable
Cosmological
tethers
The present universe
Particle
tangle;
made of
tangled
strands

Tangles Also Yield Path Integrals
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
OThe universe
OPath integrals
ODirac’s equation
OQuark generations
OLepton generations
OTight Eletron
OElectrons and
positrons
OMore diagrams
OReferences 1
OReferences 2
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 31 / 39
The strand tangle model for a ferm
i
on
i
n the path-
i
ntegral formulat
i
on
spin axis
spin axis
Step 1: The tangle
is tightened to a single
point and tethers are
neglected, yielding
a position and a phase.
Step 2: The time average
of the fluctuating point
and of its phase is taken,
yielding the wave function.
Wave function
amplitude
and central
phase
central
phase
core crossing
midpoints
Tight tangle
with phase
Spinning
electron
tangle
Spin S = 1/2
tether
flag
flag
Tethered cores follow
the Dirac equation.
Battey-Pratt & Racey, 1980.
Tight tangle cores of strands of vanishing radius are Feynman’s point particles.
Their phase (arrow / flag) rotates when advancing. Their crossing (midpoint)
density yields Dirac’s equation. (Pedagogical link.)

Spin 1/2, the Belt Trick and Dirac’s Equation
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
OThe universe
OPath integrals
ODirac’s equation
OQuark generations
OLepton generations
OTight Eletron
OElectrons and
positrons
OMore diagrams
OReferences 1
OReferences 2
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 32 / 39
Observation :
Strand
model :
spin
core
crossings
probability
densit
y
phase
time average
phase
spin
tether
Free particles (spinors) are (blurred) spinning tangle cores.
Dirac’s belt trick allows continuous (tethered) rotation (see film © by Antonio Martos).
Spin is rotation; spin value is due to strand number and tangle details.
Antiparticles are mirror tangles with opposite belt trick.
Particle momentum and energy are core wavelength and rotation frequency.
Quantum phase is 1/2 of the orientation angle of the tangle core.
The wave function is the time-averaged (“blurred”) tangle crossing density.
Maximum speed cand minimum action hold.
Strands imply the free Dirac equation iγµ∂µψ=mcψ and its propagator.
(Battey-Pratt and Racey 1980) Dirac’s equation is due to Dirac’s trick.
The principle of least action (“cosmic laziness”) is the principle of fewest
crossing switches.

6 Quarks
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
OThe universe
OPath integrals
ODirac’s equation
OQuark generations
OLepton generations
OTight Eletron
OElectrons and
positrons
OMore diagrams
OReferences 1
OReferences 2
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 33 / 39
b quark
(simplest
tangle)
The origin of the 3 quark generations  
c quark
(simplest
tangle)
s quark
(simplest
tangle)
d quark
(simplest
tangle, with
no Higgs)
t quark
(simplest
tangle)
u quark
(simplest
tangle)
one additional
crossing
d quark with
one Higgs boson
(second simplest
tangle)
u quark with
one Higgs boson
(second simplest
tangle)
one additional
crossing
one additional
crossing
one additional
crossing
one additional
crossing
one additional
crossing
one additional
crossing
d quark with
two Higgs bosons
(third simplest
tangle)
six additional
crossings
dquark
tangle
family
squark
tangle
family
uquark
tangle
family
cquark
tangle
family
bquark
tangle
family
tquark
tangle
family
etc.
Thick dashed end:
above paper plane.
Thin dashed end:
below paper plane.
No dashed end:
in paper plane.
Quarks are
infinite families
of tangles.
Each family is
due to Higgs boson
interactions.
The three dimensions
of space imply three
quark generations.

6 Leptons
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
OThe universe
OPath integrals
ODirac’s equation
OQuark generations
OLepton generations
OTight Eletron
OElectrons and
positrons
OMore diagrams
OReferences 1
OReferences 2
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 34 / 39
The origin of the
3 lepton generations  
tau &
its neutrino
(simplest tangle)
electron & its neutrino
(simplest tangle,
with no Higgs)
muon &
its neutrino
(simplest tangle)
1/3
additional
twirl
1/3
additional
twirl
el. neutrino
tangle
family
tau
neutrino
tangle
family
muon
neutrino
tangle
family
electron
tangle
family muon
tangle
family tau
tangle
family
leptons with
one Higgs boson
(second simplest
tangles)
left: neutrinos
right: charged leptons
one
additional
braid
leptons with
two Higgs bosons
(third simplest
tangles)
one
additional
braid
etc.
Also leptons are
infinite families
of tangles.
Each family is
due to Higgs boson
interactions.
The three dimensions
of space imply three
lepton generations.

The Ideal (Tight) Electron Tangle
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
OThe universe
OPath integrals
ODirac’s equation
OQuark generations
OLepton generations
OTight Eletron
OElectrons and
positrons
OMore diagrams
OReferences 1
OReferences 2
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 35 / 39
above
paper
plane
+y direction
above
+x direction
below
-y
below
paper
plane
-x
Electron tangle Q = –1, S = 1/2 0 < m << mPl
below
-z direction
spin
above
paper
plane
+y
below
-y
below
-z
below
paper
plane
-x
above
paper
plane
+z
above, +x
above
paper
plane
+z direction
According to simulations, the average shape of a flutuating tangle is the ideal,
or tight, shape.
(Link)

Electrons and Positrons
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
OThe universe
OPath integrals
ODirac’s equation
OQuark generations
OLepton generations
OTight Eletron
OElectrons and
positrons
OMore diagrams
OReferences 1
OReferences 2
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 36 / 39
t2
time average
of crossing
switches
photon
charged
fermion
e.g., electron
time
Feynman diagram
Fermion-antifermion annihilation
t1charged
antifermion
e.g., positron
electron
Q=+1
positron
Q=–1
photon
photonphoton
charged
fermion
charged
antifermion
time average
of crossing
switches
Feynman diagram
Virtual particle-antiparticle pair
t3
photon
time
t1
t2
photon
photon
photon
vacuum
vacuum
vacuum
charged
fermion
charged
antifermion
All effects
of quantum
electrodynamics
arise.
This includes
the running
of masses
and of charges.
(Link)

Additional Feynman Diagrams of QED
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
OThe universe
OPath integrals
ODirac’s equation
OQuark generations
OLepton generations
OTight Eletron
OElectrons and
positrons
OMore diagrams
OReferences 1
OReferences 2
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 37 / 39
Electron and virtual photon
photon
electron electron
time
t2
t1
t3
Interacting electron and positron
photon
electron
t2
t1
t3
positron
positron
electron
time
t2
t1
t3
electron
and photon
electron
positronelectron
positron
electron
electron
Rotation of electron cores to one loop without mass rotation Feynman diagram
Rotation of electron cores to two loops without mass rotation
plus
similar
diagrams
(Link)

Web Pages & References on Particle Physics
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
OThe universe
OPath integrals
ODirac’s equation
OQuark generations
OLepton generations
OTight Eletron
OElectrons and
positrons
OMore diagrams
OReferences 1
OReferences 2
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 38 / 39
C. Schiller, On the relation between the three Reidemeister moves and the three gauge
groups, Int. J. of Geometric Methods in Modern Physics 21 (2024) 2450057. Link.
E.P. Battey-Pratt and T.J. Racey, Geometric model for fundamental particles, International
Journal of Theoretical Physics 19 (1980) 437. Link.
C. Schiller, Testing a conjecture on quantum chromodynamics, International Journal of
Geometric Methods in Modern Physics, 20 (2023) 2350095. Link.
C. Schiller, Testing a conjecture on quantum electrodynamics, Journal of Geometry and
Physics 178 (2022) 104551. Link.
C. Schiller, Testing a conjecture on the origin of the standard model, European Physical
Journal Plus 136 (2021) 79. Link.
C. Schiller, A conjecture on deducing general relativity and the standard model with its
fundamental constants from rational tangles of strands, Physics of Particles and Nuclei 50
(2019) 259–299. Link.
Preprint: C. Schiller, Testing a model for emergent spinor wave functions explaining gauge
interactions and elementary particles.Pedagogical link.
Additional preprints at www.researchgate.net/profile/Christoph-Schiller-2/research.
Other pedagogical material at www.motionmountain.net/tangles.
Animations at www.motionmountain.net/videos.html#strands.
Experimental and theoretical predictions at www.motionmountain.net/predictions.

References on Gravity and Planck Limits
Quantum of Action
Wave Functions
Gauge interactions
Gravitation
Conclusion
Bonus Material
OThe universe
OPath integrals
ODirac’s equation
OQuark generations
OLepton generations
OTight Eletron
OElectrons and
positrons
OMore diagrams
OReferences 1
OReferences 2
Tangles of strands, particles, wave functions, Reidemeister moves, gauge groups ... www.motionmountain.net/strandsgauge.html 39 / 39
U. Hohm and C. Schiller, Testing the Minimum System Entropy and the Quantum of
Entropy, Entropy 25 (2023) 1511. Link.
C. Schiller, Testing a microscopic model for black holes deduced from maximum force,
chapter in the book „A Guide to Black Holes“, Nova Science Publishers (January 2023).
Link.
C. Schiller, Testing a conjecture on the origin of space, gravity and mass, Indian Journal
of Physics 96 (2022) 3047–3064. Link.
A. Kenath, C. Schiller and C. Sivaram, From maximum force to the field equations of
general relativity – and implications, International Journal of Modern Physics D 31 (2022)
2242019. (Honourable mention in 2022 from the Gravity Research Foundation.) Link.
C. Schiller, From maximum force to physics in 9 lines and towards relativistic quantum
gravity, Zeitschrift für Naturforschung A 78 (2022) 145–159. Link.
C. Schiller, From maximum force via the hoop conjecture to inverse square gravity,
Gravitation and Cosmology 28 (2022) 305–307. Link.
C. Schiller, Tests for maximum force and maximum power, Physical Review D 104 (2021)
124079. Link.
C. Schiller, Comment on "Maximum force and cosmic censorship", Physical Review D 104
(2021) 068501. Link.
C. Schiller, Simple derivation of minimum length, minimum dipole moment and lack of
space-time continuity, International Journal of Theoretical Physics 45 (2006) 213–227.
Link.
C. Schiller, General relativity and cosmology derived from principle of maximum power or
force, International Journal of Theoretical Physics 44 (2005) 1629–1647. Link.