Kind: captions
Language: en
most of us tie our shoelaces wrong there
are two ways to tie a knot in your
shoelaces in one you go counterclockwise
around the loop and in the other you go
clockwise
these two methods look almost identical
but one of these knots is far superior
to the other it doesn't loosen or come
untied nearly as easily to understand
why we need to delve into knot Theory
this is a whole branch of mathematics
that aims to identify categorize and
understand every possible knot that
could ever exist so far we have
discovered the first 352 million 152 252
knots each one has its own particular
properties and characteristics
I think it's fascinating that there's
something like a periodic table for
knots out there but it's not pure math
not theory has turned out to be
remarkably useful it is core to the
structure of proteins and DNA it's
leading to new materials that could be
stronger than Kevlar it's even used to
develop medicines that save millions of
lives
all of this just from trying to
understand the humble not
[Music]
so what is a knot
well in our everyday lives we see knots
like this or this but if you're trying
to rigorously study knots you want to be
able to pull them apart so you can
really see what's going on the problem
is knots like this are held together
only by tension and friction so if you
pull on them too hard they fall apart so
in order to capture the knot on the Rope
mathematicians got the idea to connect
the two ends and now well you can tease
the knot apart to study it but it will
never fundamentally change so in not
Theory all knots exist on closed Loops
this means the simplest knot you can
have is just a circle like this now
admittedly this is not much of a knot
which is why it is called an unknot
[Music]
here is another knot again it's made of
a single piece of rope that forms a
closed loop here is another
two knots are only different if you
can't make one into the other without
breaking the loop this is the simplest
knot after the unknot it is called the
trefoil and you can see that there's no
way for me to turn this back into a
circle unless I actually break it open
take out the knot and then close it up
again
now I have two unknots
it is surprisingly hard to tell two
knots apart by eye here is a simple
mystery knot is it an unknot a trefoil
or neither I'll give you a second to
figure it out
it is in fact a trefoil which you can
see if I just untwist this and rearrange
the knot a little bit
and our first complicated knot
well it is actually just the unknot I'm
going to try to disentangle it so you
can see that
there you go
it was just a single Loop
in fact these are all unknots and this
is where the problem begins you can't
just randomly tangle some rope and
connect the ends to make a new
mathematical knot you need to prove it's
not just a tangled up version of another
knot so how do you tell two knots apart
this one question also known as the not
equivalence problem is so famously
difficult that it's propelled the entire
field of not theory for over 150 years
Alan Turing even wrote in his final
publication no systematic method is yet
known by which one can tell whether two
knots are the same a decision problem
which might well be unsolvable is the
one concerning knots the results in this
article set certain bounds to what we
can hope to achieve purely by reasoning
previously the most famous not problem
in history was the gordian knot it was
said that whoever untangled this massive
knotted rope was destined to rule all of
Asia
Legend goes Alexander the Great simply
came along and sliced right through it
that would not be a valid solution in
not Theory
and there are other famous knots of
History the endless knot is seen as far
back as clay tablets from the Indus
Valley and it was used in medieval
Celtic designs Chinese Network and
Hinduism and Buddhism
in Incan civilization knots were tied on
chords called kipu to track everything
from taxes to calendars
you can even find a knot in the coat of
arms for the house of boromeo an Italian
Noble family that has existed since the
1300s the boroman Rings are technically
a link which is just a not with multiple
Loops of rope the most basic link is the
unlink two Loops which aren't actually
connected much like the unknot after
that is the Huff link then later the
boroman Rings and more
but the not equivalence problem was only
encountered centuries later
[Music]
in January of 1867 Scottish physicist
Peter Guthrie Tate showed off his
homemade smoke machine to renowned
scientist William Thompson later Lord
Kelvin Tate had read a paper that said a
Vortex ring should be eternally stable
in an ideal fluid
so intrigued he set up two wooden boxes
containing a kind of toxic mixture of
ammonia sulfuric acid and salt when he
tapped a towel stretched across the back
of each box the chemical smoke popped
out of a circular cutout in perfect
rings
Kelvin watching was transfixed by the
Rings he had been pondering the
composition of atoms a fundamental
question of the time and suddenly he saw
an answer he declared that atoms must be
made out of Vortex rings of ether an
invisible everywhere medium different
knots of Vortex Rings would make
different elements the shape of the hot
Flink explained the double spectral
lines of sodium the simple unknot ring
was hydrogen
Tate was skeptical but as kelvin's
Vortex model of the atom became a
leading theory Tate began investigating
knots in Earnest in his mind creating a
periodic table of the elements with
every new knot he found
The Crossing number is an easy way to
categorize knots just take the simplest
form of a knot one with no extra twists
or Tangles and count up all its
Crossings
by hand Tate discovered a three Crossing
knot the trefoil then a four Crossing
knot the figure eight then two five
Crossing knots three six Crossing knots
and seven seven crossing knots
one quick note knots are additive you
can stick multiple knots together to
make a new knot like combining these two
trefoils into a six Crossing knot this
is called a composite knot but some
knots aren't decomposable into simpler
knots these are known as Prime knots
since all composite knots are just built
from primes people mainly focus on
tabulating Prime knots
unfortunately for Tate warning signs
were on the horizon for Lord kelvin's
Vortex theory of the atom mendeleev's
first periodic table had been published
in 1869 the Michelson Morley experiment
sowed seeds of doubt about an ether in
1887 and the most damning result was JJ
Thompson's discovery of the electron in
1897 so there were particles smaller
than the atom which were inside atoms
but Tate was already in too deep with
knots to stop he had even roped in his
academic rival and close friend James
Clerk Maxwell of Maxwell's equations
vein thanks to Tate's influence Maxwell
became a not Enthusiast for the rest of
his life
his last poem written shortly before his
death from stomach cancer even begins my
Souls an amphi chiral knot upon a liquid
Vortex rot
[Music]
assisted in part by hundreds of letters
with Maxwell Tate published his list of
knots up to seven Crossings in 1877. the
first map paper with the word knots in
its title
then Tate paused his search for seven
years in a speech he stated the
requisite labor increases with extreme
rapidity as the number of Crossings is
increased someone with the requisite
Leisure should try to extend this list
if possible up to 11. two mathematicians
took him up on his call for help Thomas
Kirkman and Charles little
together the three of them were able to
find all 21 8 Crossing knots all 49 9
Crossing knots and all 166 10 Crossing
knots by 18.99 just two years before
Tate's death
was all done painstakingly by hand Tate
admits in his paper I cannot be
absolutely certain that all those groups
are essentially different one from
another
in the process of tabulating knots he
had uncovered the central problem of how
to possibly tell them apart
but miraculously take Kirkman and little
had done a near perfect job with their
knot tables their list stood for 75
years without change until a single
correction in 1973 but more on that
later
for decades after Tate's death little
progress was made on the not equivalence
problem but in 1927 German mathematician
Kurt reitemeister proved a radical
theorem you only need three types of
moves to transform any two identical
knots into each other the twist the Poke
and the slide where you move a string
from one side of a Crossing to the other
now we can prove some knots are the same
if you can show that they're connected
by rydammeister moves
[Music]
you've proven that they must be
identical
but we still don't know how to prove any
knots different from each other because
you could do ridermeister moves on one
knot for centuries without it ever
looking like the other knot and maybe
they're actually different but maybe
they're the same and you just never made
the right move to show that that is true
this may have been where Turing was
coming from when he called the not
equivalence problem potentially
undecidable
but
in 1961 mathematician Wolfgang Hawkins
created a computer algorithm that solved
the not equivalence problem definitively
for this specific case of distinguishing
any knot from the unknot that said his
paper was over 130 pages long and the
algorithm would have taken longer than
the age of the universe to run for large
knots
in 2001 building on Hawkins work
mathematicians found a way to
distinguish between any knot and the
unknot by simply setting an upper bound
on the number of randomized removes
needed to connect them if you check all
sequences of randomized removes up to
that number you can prove if the knot is
the unknot or not
there's just one problem that upper
bound was 2 to the 100 billion n moves
as of today the upper bound has improved
dramatically to just 236 n to the power
of 11. now while smaller than before
checking all possible sequences of
randomized removes up to this number is
still unfathomable for a single Crossing
knot this is larger than the number of
stars in the observable universe
in 2011 mathematicians found an upper
bound on the number of ridermeister
moves needed to connect any two knots or
links solving the entire not equivalence
problem
this is the upper bound
first raise two to the second power then
raise that to the power of two again
this operation is called titration and
it grows fast
now keep doing it until you have raised
2 to itself 10 to the million n times
cap it off with n again
this is easily the largest number we
have ever shown in a video
but even just to have a solution is
remarkable given that Turing thought the
problem was potentially undecidable only
60 years earlier
if it's this hard to tell two knots
apart how have we managed to tabulate
350 million different knots well there
are some properties of a knot that never
change no matter how much you twist or
tangle it up these are called invariants
and these invariants will be different
for some knots compared to other ones so
you can use them as Hallmarks of a
particular knot they're not perfectly
discriminating I mean some knots will
share in variants but if two knots have
different invariants then you know for
sure that they are different
Crossing number is itself an invariant
two knots can't be identical if they
have different Crossing numbers
but the Crossing number is surprisingly
difficult to calculate you can put extra
Crossings into any knot like just throw
in a bunch of twists different
variations of the same knot are known as
different projections of that knot
Crossing number measures the least
number of Crossings a knot can have but
it only works for the simplest
projection of a knot also known as its
reduced form but it's difficult to
ensure that a knot is fully reduced
instead we can use another invariant one
that's true right away for all
projections of a knot so it'll give the
same value both for a messy trefoil and
for a reduced trefoil
this first invariant is tri-colorability
or whether or not can be colored in with
three colors
take a diagram of a knot and color in
each individual segment these are just
separated by under Crossings where you
would lift your pen off the page
tri-colorability only has two rules
first you must use at least two colors
because you can color any knot in with
one color and second at Crossings the
three intersecting strands must either
be all the same color or all different
colors basically no two colored
Crossings
there are just two categories of this
invariant either a knot is tri-colorable
or it's not identical knots must match
so if one knot is tri-colorable and the
other one isn't then you know they're
different knots it's hard to believe
that tri-colorability is constant across
any possible projection of the same knot
but since you only need rhitomized
removes to move between projections we
just need to prove that it isn't
affected by ridermeister moves the twist
is easy everything's one color already
and it stays that way with the Poke the
intersection of two colors means that
the loop formed must become the third
color so we have three colors at every
intersection with the slide you never
have to break tri-colorability because
you start with three colors at three
intersections and then switch one
intersection to just one color
so any knot will maintain its
tri-colorability no matter what
ridermeister moves you do
this is a good time to note that we
never actually proved the trefoil and
the unknot were two different nods but
we can do it now with tri-colorability
the unknot is not tri-colorable since
you can't use at least two colors to
color it in and the trefoil is easily
tri-colorable just color in each of the
three segments a different color The
Crossings all have three colors so it is
tri-colorable
now we know every possible projection of
the trefoil is tri-colorable while every
possible projection of the unknot isn't
so these two knots must be different
knots
this invariant isn't very specific it
only gives you two categories across all
nods in fact the next not after the
trefoil the figure eight knot isn't
tri-colorable there's always a Crossing
with two colors so how do we prove that
this is different from the unknot which
also isn't tri-colorable
tri-colorability expands into a much
more powerful invariant called P color
ability where P can be any prime number
besides two instead of using colors
we'll number each strand with integers
between 0 and P minus one
P color ability has two rules first you
must use at least two different numbers
second at Crossings the two bottom
strands added together and divided by P
must give the same remainder as twice
the top strand divided by P
tri-colorability was just a simple
version of this
if we go from three to five color
ability for the figure eight knot we can
number the strands zero one
and then this strand must give a
remainder of zero so four
and this strand must give a remainder of
two so three
this knot is five colorable so it's not
the I'm not
P color ability is a huge tool the
unknot is completely uncollarable so any
knot with any color ability can't be the
unknot
P color ability still doesn't cover
everything some of the most powerful
invariants right now the ones that can
distinguish between the most unique
knots are polynomials the Alexander
polynomial was the first one discovered
back in 1923 before even ridermeister
moves like P color ability it relies on
only two rules
the first is that the Alexander
polynomial of the unnaught is equal to
one the second is that you can zoom in
on any single Crossing of a knot and
Vary it in three possible positions
forward backward and separate the
Alexander polynomial gives a
relationship between the three resulting
knots
Let's do an example what's the Alexander
polynomial for the unlink well if we
zoom into this separate Crossing and
then vary it we see that the other two
knots formed are both the unknot so we
can plug in one for both of them and we
get that the Alexander polynomial for
the unlink must be zero
then we can do the same for the hot
Flink taking this Crossing as the
forward Crossing then seeing that the
backward Crossing gives us the unlink
and the separate Crossing is the unknot
so the Alexander polynomial for the Hop
link is minus t to the half plus t to
the minus a half
and now we can do the trefoil when we
vary this Crossing we get that the
backward Crossing gives us the unknot
and the separate Crossing gives the Huff
link so the Alexander polynomial is T
minus one plus t to the negative one
the polynomial is designed so that we
get separate results for as many knots
and links as we can and this is
recursive we can calculate the
polynomial forever for bigger and bigger
knots
the Alexander polynomial stood unchanged
for over 60 years as the not invariant
of choice but in 1984 it was upended by
an unlikely Discovery mathematician Von
Jones had been working on a type of
algebra for statistical mechanics a
concept in physics when he realized his
work resembled a series of equations in
not Theory
he traveled to New York to consult not
theorist Joan Berman at Columbia
University who helped refine his
equations into a not invariant they met
again a week later and tested it against
knot diagrams from Berman's filing
cabinet quickly realizing Jones had
discovered a brand new polynomial
invariant
he scribbled down all their work in a
15-page letter
the Jones polynomial is like the
Alexander but with the more specific
equation for the second rule that lets
it distinguish many more knots for this
discovery Jones won the fields medal in
1990.
the First new polynomial invariant
kicked up a fervor in not Theory just
months after Jones result six
mathematicians each independently found
an improved version of his polynomial
with two variables instead of one
the editors of the American math Society
published all their papers together
naming it the humfly polynomial two
polish mathematicians missed the news
and discovered it again a couple months
later upon which it became the hum
flight polynomial
[Music]
none of these invariance works alone
just like if you were searching for a
person you'd start by checking a first
name then a last name then a birthday
and so on to eventually narrow your
search down to just one person similarly
knots have dozens of invariants which
when taken together uniquely identify
them with invariance to prove if knots
are different and write a Meister moves
to prove if not so the same you can
attack from two angles to meet the
gargantuan task of distinguishing every
single knot
but this method isn't perfect these two
knots were listed next to each other in
Tates not tables for over 75 years they
were the Same by all invariant accounts
so Tate and little likely tried
rhydamized removes to see if they could
transform one into the other and once
they failed they listed them as two
separate knots
Kenneth perco a lawyer who had studied
not Theory spotted them in 1973 while
looking through Little's table of 10
Crossing knots suspicious of their
similarities he pulled out a yellow
legal pad to sketch some ridermeister
moves and he quickly found a way to
connect the two knots these two
projections now known as the perco pair
are the same not so the knot tables of
tape Kirkman and little were issued
their single correction and instead of
166 10 Crossing knots there are
165.
it had taken Decades of work to tabulate
all 249 Prime knots up to 10 Crossings
now when dared tackled the 11 Crossings
until John Conway he found all 552 and
claimed he did it in a single afternoon
this was the last tabulation by hand in
the 80s dowker and thistlethwait built a
computer algorithm to count all 12 and
13 Crossing knots
thistlethwait later joined forces with
haustin weeks to tabulate all 14 15 and
16 Crossing knots in a paper titled the
first one million 701 936 knots
the method they used is still the one
used today employing a computer to list
all possible knots and then using
invariants to weed out duplicates they
split into two teams and cross-checked
their results aligning perfectly on all
but four knots on their first try
in 2020 mathematician Ben burton
single-handedly tabulated all 17 18 and
19 Crossing knots bringing the total
number of known prime knots to 352
million 152
252. his project was so computationally
intensive several hundred computers had
to run for months before obtaining the
final number
the hardest part of not tabulation is
counting up every knot and then
carefully eliminating duplicates but if
you just want to generate a huge number
of distinct knots you can make
alternating knots knots with crossings
that alternate over under this
computation is much easier though it
leaves out most knots and back in 2007
this method was used to find alternating
knots up to an absurd 24 Crossings so in
total we know of 159 billion 965 million
353 knots
of course we're missing a lot in between
there
not theory was always just pure math all
the algorithms invariants and
tabulations were knowledge for the sake
of knowledge but in 1989 chemist
jean-pierrezovaj tied molecules around
Copper ions to form the first ever
synthetic knotted molecule this trefoil
knot restricted the atoms from unfurling
trapping them in higher energy states to
give the molecule new properties any
type of knot tied in a molecule will
change its properties and we know of
over 159 billion knots so if you can tie
a molecule into each of those knots
that's 159 billion new unique materials
created from a single molecule
though after the trefoil chemists have
only managed to tie five other molecular
knots to date it's a difficult task
since they can't just nudge individual
ions into place molecules must be built
to self-assemble into knots knot Theory
helps identify which knots match
available molecular templates symmetric
knots are easier for one and how to
arrange those to assemble the knot
the most complex knot yet created is the
819 knot with 192 atoms tied around a
central chloride ion
this molecule holds the Guinness World
Record for tightest knot in the world
defined as the most Crossings per unit
length in this case eight Crossings in
20 nanometers since it's knotted around
a chloride ion once the ion is removed
this molecule is one of the strongest
chloride binders in existence
the field is still new for specific
applications chemists are just focused
on creating molecular knots before
thinking about materials development but
they hope to eventually build things
like durable Fabrics stronger than
Kevlar
knot theory is also critical to
biological processes that have saved
millions of lives
bacterial DNA consists of a single Loop
of the double helix molecule this shape
means that it always forms a knotted
link when it replicates the bacteria
can't separate into two cells with their
DNA tangled up like this so they have an
enzyme called type 2 topoisomerase which
Snips and reconnects the DNA this turns
their link DNA back into an unlink so
they can replicate cleanly if you
inhibit type 2 topiosomerase the
bacteria can't replicate properly and in
fact they die this is how some of the
most common antibiotics in the world
called quinolones operate
human DNA while not circular is long
enough to also get into Tangles each
cell in your body contains two meters of
DNA that's the equivalent of stuffing
200 kilometers of fishing line into a
basketball when this mess inevitably
Tangles human type 2 topoisomerases come
to make Crossing changes the human
version of the enzyme is different
enough from the bacterial version that
it's unaffected by antibiotics
but human topoisomerases are sometimes
intentionally inhibited this stops
replication and kills cells
predominantly the rapidly dividing
cancer cells so it's one of the most
common forms of chemotherapy
[Music]
biologists needed knot Theory to First
understand the mechanism of type 2
topoisomerase once they observed it was
decreasing the Crossing number of knots
in DNA two at a time they realized it
had to be cutting and rejoining entire
double strands of DNA and there are many
other obscure topoisomerases that act on
DNA knot theory is used to analyze the
knots they tie or untie and how they
operate as a result
it's not just DNA that knots one percent
of all proteins have various knots in
their fundamental structure if they get
misnotted they malfunction so being able
to accurately tell not to part helps
understand these proteins mechanisms as
well as how to potentially repair or
utilize them
when it comes to your shoelaces both of
the common ways to tie the knot are
composed of two trefoils on top of each
other I'm going to tie some rope around
my leg to make this easier to see when
you go counterclockwise around the loop
well then you form two identical
trefoils on top of each other this is
also known as a granny knot but when you
go clockwise around the loop then you
get mirror imaged trefoils on top of
each other this is also known as a
square knot and it doesn't loosen as
easily so we should all be tying our
shoelaces like this clockwise around the
loop
most of us aren't I mean I'm I'm not
usually I normally do it like this
a simple overhand knot is just the
trefoil the bowline knot the most common
knot for boating or just holding things
together is the six two knot and any
knot tied without using the ends also
known as in the bite that is just an a
nut so a slip knot is an example
of an unknot
in 2007 researchers Dorian Ramer and
Douglas Smith conducted
3415 Trials of spinning string in boxes
to study how knots form in the real
world
[Music]
they ended up creating 120 different
types of knots some as complicated as 11
Crossings
they found that longer agitation time
led to a higher chance of nodding longer
string did as well except this
probability decreased once the string
was put in a smaller box which
restrained its motion
so if you want to keep something like
headphones from nodding in your pocket
where you can't adjust string length or
agitation time then your best bet is to
confine them to as small a space as
possible
Raymer and Smith also proposed a model
for real world knot formation a series
of Loops are first formed when a string
is placed into a container
then when it's agitated a free end of
the string gets woven up and down
through the loops braiding itself into
them to form knots
[Music]
foreign
let's see
uh
a nut
so coiling up your wires is actually
setting yourself up for failure because
you're forming a bunch of Loops for a
loose end to braid perfectly into a knot
so instead what you want to do is
restrict its movement whether by using a
small box or increasing string stiffness
DNA increases its stiffness by super
coiling you can do the same with your
wires I just double it up like this and
then I twist from the middle and this is
going to stiffen the length of wire now
this is naturally going to want to sort
of coil in on itself and it's going to
look like a big Tangled mess but all you
have to do is take the opposite ends of
the headphone and pull apart
and there's no Tangles no knots
their study won an IG Nobel Prize it has
been cited in studies of knots in
surgical catheters and even linked to an
Apple patent for stiffer earbud wires
not Theory began as a failed Theory of
Everything and for the next Century it
was a standalone field of math propelled
by nothing more than intellectual
curiosity but in recent years it's
reclaimed its original potential today
not theory is a theory of everything
from headphone Tangles to Material
Science to chemotherapy
in 1889 Calvin gave a presidential
address to the British institution of
electrical engineers about his failed
atomic theory of knots
I am afraid I must end by saying that
the difficulties are so great in the way
of forming anything like a comprehensive
theory that we cannot even imagine a
finger post pointing to a way that leads
us towards the explanation
but this time next year this time 10
years this time 100 years I cannot doubt
but that these things which now seem to
us so mysterious will be no Mysteries at
all that the scales will fall from our
eyes that we shall learn to look on
things in a different way when that
which is now a difficulty will be the
only common sense and intelligible way
of looking at the subject
thank you
knot theory is a perfect example of how
knowledge in one area can become a tool
to understand countless others from
learning how to tell a trefoil apart
from an unknot not theorists have built
all the way up to discovering brand new
proteins if you want a quick and easy
way to build out your own mental toolkit
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