Tech note: All figures herewithin are produced in SVG. If you cannot see
them, I recommend you figure out how to view SVGs in your browser.
A few weeks ago, my friend radii was puzzling in his server hall. He
asked if it was possible to prove that what he wanted to do was
impossible, or if he had to remain with his gut feeling. I asked him,
and got the following explanation:
He had two strands of something ropelike, both fixed at large
furnishings at one end, and fixed in a fixed sized loop at the other.
He wanted to take these, and link them fast to each other in this
I started thinking about the problem, and am now convinced I can prove
the impossibility he asked for by basic techniques of knot theory. The
argument is what I'll fill this blog post about.
First observation is that due to the size of the fixtures, we can
essentially consider the endpoints fixed. Not only that, but since we
cannot thread the loops over the fixtures, there's no way to just stick
that loose end through any of the loops. So we can basically extract a
cube of space and require that all of our modifications be contained
completely within this cube, and then stick the endpoints just outside
This is something called a framed knot, and a popular object of study.
My argument is going to manage to steer basically entirely clear of
this, though, and I'll just mention it as an extra property to remember.
The second observation is that the size of the loops is basically
irrelevant. So we can make the length of the single strand part as
small as we like, and as close to the fixed end as we like. Hence, for
all purposes, we essentially want to produce a link like this:
And this is essentially the problem I will approach. Now, I'm going to
prove the impossibility by a method similar to how we prove that the
unknot are different; using
Equality of knots is basically equivalent to being able to go from the
drawing of one knot to the drawing of another knot by the Reidemeister
moves. I warmly
recommend the wikipedia page for a first glance - these turn out to be
absolutely central for constructing knot invariants.
For the trefoil argument, we start with discussing coloration of
knots. Given three colors: red, green and blue, we may color each
strand in a knot diagram with one of the colors. We require that at
every crossing, each strand is either the same color, or different
color. Obviously, every knot, including the unknot, can be colored in
all the same color. And it turns out that the Reidemeister moves are
compatible with knot coloration. The proof of this is entirely
pictorial; up to a permutation of the colors, the following are all
the options we have:
The take home message is that for every crossing configuration, with
fixed frame outside, all Reidemeister moves respect the colors at the
So, how may we apply this to our problem? We still have the
characterization of knot equality by finding sequences of Reidemeister
moves. So we'll start with the unknotted version of radii's problem,
and a chosen coloration:
Notice that in the diagrams above, the only way to introduce a new color
is to already have at least two colors present. Hence, any Reidemeister
moves made on this untied configuration will stay entirely green.
However, the target configuration does have this coloration:
Notice that the fixture coloration is identical to the one in the
unknotted version, but that this version has all colors represented.
However, since the Reidemeister moves cannot possibly introduce a new
color starting in the untied version, they can never arrive at this