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There now follows a large brain dump concerning knot theory...
At one time, it was briefly theorised that maybe the ~105 elements of
the periodic table were each a little tangle of energy, and different
kinds of tangling gave rise to different chemical properties. This
sparked a great deal of interest in knot theory. Later this idea was
abandoned, and knot theory became unpopular again. But some people still
study it.
(With String Theory, the idea seems to be coming back somewhat. But I
digress...)
So what actually *is* knot theory? Well, it's the study of mathematical
"knots". As you might expect, these abstract entities have properties
similar to but not quite the same as a real knot in a piece of string.
Fundamentally, a "knot" is a *closed loop* of infinitely bendy, stretchy
string. Notice that if you have the free ends of a piece of string, you
can (in principle) always untangle it given sufficient patience. A
mathematical knot has not free ends, and hence cannot be untangled.
That's what makes it a well-defined entity that you can sit down and study.
Two tangled up bits of this hypothetical string are considered "the
same" if you can turn one tangle into the other without cutting the
string or allowing it to pass through itself. (Shrinking parts of the
knot down to 0 size to make them dissappear is also cheating.)
It is a basic result of knot theory that it's often insanely hard to
determine whether or not two given knots are actually the same or not.
The simplest knot is just a circle. This is called the "trivial knot" or
"unknot", since it's completely untangled. Notice that if you take an
elastic band, you can tangle it up really good without actually cutting
it; this is *still* considered isomorphic to the trivial knot. The next
up is the trefoil knot. Again, this can have several appearences,
depending on how you arrange it.
It is useful to think of "the knot", which is an invariant, unchanging
thing, and "projections" of the knot. A knot projection is basically a
2D drawing of a particular configuration of the knot. Any given knot has
an infinite number of possible projections, ranging from the simple to
the utterly convoluted.
Aside from stretching and shrinking the string, a knot projection can be
"changed" in three fundamental ways. These ways are named after a
mathematician who's name is beyond my ability to spell or pronounce.
Remember we're talking about knot projections, that is, 2D drawings of a
particular configuration of a knot.
- A type I move involves taking a strand, making a small loop with it,
and poking it over the top of another strand. (Or, alternatively, under
it.) This makes the projection slightly more complicated (there are now
more crossings), but does not alter the knot itself. The reverse
process, i.e., drawing a loop back over (or under) a strand, simplifies
the projection. (Note that a loop *around* a strand cannot be
so-simplified. Only a loop completely over or completely under.)
- A type II move involves taking a straight strand and twisting it to
form a loop. You have now added one new crossing to the projection.
Alternatively, untwisting the strand to eliminate a loop, thus reducing
the number of crossings. (You can only do this if nothing passes through
the loop, of course.)
- A type III move involves moving a strand from one side of an unrelated
crossing to the other.
These three moves, then, alter the 2D projection of a knot without
actually changing its fundamental structure --- i.e., which knot it is.
One of the ways to attempt to classify knots is by the construction of
"knot invariants". A knot invariant is basically something you can
compute from the projection of a knot, and which is the same for *all*
possible projections of a given knot.
An immediate consequence of this is that if two projections yield
different values for a given invariant, they are provably *not* the same
knot. (However, if they yield the same value, this does not prove that
they *are* the same; that would be too easy. It is however a desirable
property that it should be "unlikely" for different knots to yield the
same value.)
The way to construct an invariant, of course, is to prove that all of
the three moves listed above leave the invariant unchanged.
The details of specific invariants escape me now. Suffice it to say that
several mathematicians have come up with polynomials that can be
constructed from a knot projection. The polynomial itself doesn't
calculate anything interesting, but its formula is derived from the knot
projection. The algebraic properties of polynomials are used in such a
way that the three moves change the subterms, but when you simplify it
all into standard polynomial form, you get the same answer.
I remember one of these polynomials involves recursively splitting the
knot projection into simpler ones, eventually building a trivial
polynomial for each part, and then combining these polynomials back
together according to how the original parts where connected. Shift some
algebra, and at the end every projection of a given knot comes up with
the exact same polynomial.
There are invariants which are *not* polynomials. It's simply that most
of the popular invariants happen to be polynomials.
Invariants are a nice way to compare knot projections in an attempt to
see if they are different. But how to *describe* a knot unambiguously?
Well, several methods have been put forward.
One rather entertaining way goes something like this. (I've probably
screwed up the algorithm; this is from memory.)
- Pick a starting point on the string, and draw an arrow representing a
direction. Doesn't matter what you pick, but stick to it.
- Trace your way around the knot. Each time you reach a crossing, number
it, starting from 1. If the strand you're on goes over the top, use a
positive number. If it goes under, assign a negative number.
- Write down a list of all the pairs of numbers at each crossing.
- Throw away the lowest number in each pair (ignoring sign).
It is possible to completely reconstruct the know from the list you're
left with. As an example, I just tried a figure of eight knot. From
this, I get the list
-1, +4
+2, -7
-3, +6
-5, +8
Throwing away half the numbers, we get +4, -7, +6, +8, in that order.
To reconstruct the knot, we need to fill in the table:
??, +4
??, -7
??, +6
??, +8
Since the *first* column is always in ascending order, and each pair
always contains one + and one -, we can reconstruct quite easily. Once
we have the two-column table, you can imagine taking some bendy plastic
tubing with some crossings taped together and connecting the crossings
in ascending order to reproduce the knot. (1 is connected to 2, 2 is
connected to 3...)
This method, of course, just describes a specific projection of a knot.
There are many ways to project the same knot, and hence many possible
sequences of number to describe it. (Changing the starting point also
alters all the numbers.)
An alternative way to describe knots is by "braid theory".
A "braid" is a series of vertical strands. Initially, they are all
parallel. If you say "+3", that means that strand 3 and strand 4 swap
places, with strand 3 going over the top of strand 4. Alternatively,
"-3" means the same swap, but strand 4 going over the top.
In this way, you can say "-3, +5, +2". This describes a sequence of
strand swaps, starting from the top and working downwards. Something
like this:
1 2 3 4 5 6
| | | | | |
| | \ / | |
| | / | |
| | / \ | |
| | | | | |
| | | | \ /
| | | | \
| | | | / \
| | | | | |
| \ / | | |
| \ | | |
| / \ | | |
| | | | | |
1 2 3 4 5 6
So that's a braid. Now if you imagine taking this and bending it over so
that the ends at the top connect with the ends at the bottom, this would
make a closed loop. In fact, in this case, the result would be *several*
closed loops. The 1 strand would be an unknot, not connected to anything
else. Strands 5 and 6 would become a single strand, which can then be
untwizzled to make an unknot. And strands 2, 3 and 4 would be connected;
off the top of my head, I'm not sure if this would be a nontrivial knot.
Such a collection of possibly-connected knots is called a "link". In
general, "closing" a braid (i.e., connecting its ends together) produces
a link. Sometimes the whole link consists of one knot (i.e., one
continuous strand), and sometimes several knots that can be seperated.
And occasionally, several knots connected such that you can't seperate
them without cutting.
The fun part, of course, is the algebraic structure of a braid.
Sometimes when you move a twist up or down the sequence, it changes the
resulting link when the braid is closed. And sometimes it doesn't.
Teasing how the relationships for this can get quite interesting.
Best of all: any possible knot or link can be represented as a braid.
(Although working out how usually isn't easy.)
Another method for constructing knots is to use "tangles".
A "tangle" is a section of string or strings that have 4 ends. The ends
are locked in place and can't move, and you can't loop the strands over
those ends. If you imagine drawing a square with one end bolted to each
corner of the square and the strands inside not allowed to leav the
confines of the square, that's roughly what a tangle is.
Again we have an algebra of tangle construction here. (I may well be
getting some lefts/rights mixed up here, but the ideas are essentially
correct.)
We start with the "0 tangle". This is where the two top corners are
linked, and the two bottom corners are linked, and the strands aren't
tangled up in any way.
Then we have the "1 tangle". This is where you take the 0 tangle and
swap round the two right ends, such that the strand from the bottom-left
corner passes over the one from the top-left corner. The "-1 tangle" is
identical, but twisted the opposite way. (I.e., the top-left thread is
on top.)
You can "add" two tangles together by placing them side by side, and
connecting the two right-hand ends of the left tangle to the two
left-hand ends of the right tangle.
If, for example, you add a 0 tangle to a 0 tangle, you get a new 0
tangle. If you add a 1 tangle to a -1 tangle, you also get... a tangle
where one thread moves over the other, and then back again. Performing a
type-I move, this becomes the 0 tangle again.
So, 0 + 0 = 0 and (+1) + (-1) = 0. That's cute. But if you add a 1
tangle to a 2 tangle, you get a tangle where the two threads cross over
each other twice in the same direction - the "2 tangle". (A "-2 tangle"
is defined similarly, but with the twist in the opposite direction.)
So, an N tangle is the 0 tangle twisted N times clockwise, and a -N
tangle is twisted N times anticlockwise. (Assuming you look at it from
the right direction.)
There is also an "infinity tangle", which is like the 0 tangle, rotated
This involves placing one above the over, and joining the corners in
that direction instead.
Here, however, we find that tangle algebra doesn't work *quite* like
number arithmetic; if you multiply the 0 tangle by the 0 tangle, you get
something that isn't even a tangle; it's like a 0 tangle with a trivial
knot floating in the middle of it. And a 1 tangle multiplied by a 1
tangle gives you something that can't be described any other way. (In
particular, *not* a 1 tangle!)
By pairing up the ends of a tangle (either vertically or horizontally -
it makes a difference) you can again construct any possible knot. And
again there's an interesting algebra of operations which are equivilent
and those which are not.
In a similar way, you can "add" regular knots. You take two knots, cut
them both, and join the cut parts. The thing is, depending on exactly
where you cut them, and how you join the ends up, you can make several
different knots in any addition operation. So for general knots,
"adding" isn't very precisely defined. (At least, if you just specify
two knots and that they be added, the result is not well-defined. You
need to specify lots of extra info to make it well-defined. Even the
knot projection might make a difference.)
There are other ways of making knots too. The trefoil is a "toriodal
knot". That is, you can generate it by marking a point on a circle, and
rotating that circle while sweeping it around a perpendicular circle.
(In other words, tracing a path on the surface of a torus.) By varying
the number of rotations of one circle for each rotation of the other,
you can build various different knots, of which the trefoil is just one.
(There is an infinite set of parameters that generate any given toriodal
knot, however.)
Related to this are "cable knots". This involves sweeping a circle not
along a circle but along another knot. Sometimes the result can be
unravelled to be isomorphic to the original knot; sometimes it can't.
(In other words, sometimes it's a genuinely new knot.) You can also
generate links this way by not rotating the circle as it is swept;
sometimes these links are seperable, and sometimes they aren't.
What about knots in 4D instead of just 3D? What would that be like?
Well, it turns out to be pretty boring, actually. In 4D, and knot
composed of a 1D strand can actually be completely untangled. That is,
every knot is equivilent to the trivial knot in 4D. That's not terribly
interesting.
What you *can* do, however, is construct knots out of a 2D "ribbon"
rather than a 1D "string". The result is a family of knots that only
exist in 4D, but it's really *far* too mind-bending to think about.
(Projecting back into 3D can look pretty though...)
While we're on the subject of pretty pictures, there is various software
to perform "knot relaxation". The idea is to take a description of a
knot, and try to "simplify" it by treating the knot as being made out of
bendy, stretchy plastic and doing a small physical simulation. An
intricately tangled length of string is likely to untangle itself given
sufficient energy to escape any local minima and assume a low-energy
final state.
It can be quite interesting to watch, and it's sometimes a useful way to
figure out if two knots are the same. If they are, they typically tend
to assume similar final configurations. (But not always...)
This has been another broadcast brought to you by an under-employed
computer science graduate, for the benefit of similarly over-interested
souls. TTFN!
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