Hi, I'm Henry Segerman, and these both look like the trefoil knot:
the simplest knot there is other than the unknot, which is just a circle.
However, if you've read the title of this video,
you'll suspect that one of these is not what it seems.
Well, this one isn't actually a trefoil.
This strand here looks like
it goes underneath and connects up with this strand over here.
But in fact, it doesn't.
What actually happens
is that it does a hairpin turn backwards and heads off in some other direction.
If this was made from a more flexible material you could take this hairpin turn
and thread it back through this loop here, and the whole thing would unravel
into just an unknotted loop - this is an unknot.
The only way you'd be able to tell that something is wrong
is that the knot slopes back too quickly as it approaches this crossing,
in comparison to this one here.
This one isn't a trefoil either.
In fact, it's one big hairpin loop tied in a knot.
You could untie it and open it out. And again, it would be a circle.
This is also the unknot.
This one is actually the trefoil, no special camera angles required.
The other two both look like trefoils,
but they're actually unknots. They're unknots "disguised" as trefoils.
What about the other direction though?
Could you have a trefoil that's disguised as an unknot?
Well, here's an example.
I've got this mirror here
so you can see the side view of what's going on with this knot.
We can't do this with the simplest possible
diagram of an unknot, which would just be a circle.
But if you've got just one crossing, that's enough to hide the necessary trickery.
In the mirror you can see there's the
over, under, over, under, over, under.
Yeah, this is a trefoil.
Here's a question for you: Could you disguise any knot as the unknot
like this, or is there something special about the trefoil?
If you had some other knot and you were going to try and do this,
how would you do it?
Or like the examples from before - there we were disguising the unknot
as the trefoil. Could you disguise some other knot as the trefoil?
Well, let's see how these disguises work.
A knot is an interesting thing to try to disguise,
because most of the time you can see almost all of it from any direction.
The only places you can hide things are directly in the shadow
behind what you're supposed to see.
But that's actually quite a lot of space to play with.
You can do whatever you want back there
as long as you have the correct strands at the front.
Where those strands appear to go under each other, you can instead
do something else - either doing a hairpin turn back under the incoming strand
or turning and heading off under the overstrand.
There's still something to figure out, though.
If you wanted to disguise some super complicated knot,
would you be able to fit it into this shadow?
Let me know what you think in the comments.
Beyond the question of where the knot goes in the shadow,
making it look good required some thought.
We need to be very fluid in the depth of features
so that the curves can look nice and smooth. At the same time
the design must stay precisely behind the disguise at the front.
So I wanted to be able to make lots of little "by-hand" tweaks to the depth
of parts of the knot, while not accidentally straying out of the shadow.
Here's how it works.
I've got this long curve which wiggles up and down.
I wrote code that takes this curve and wraps it around
the precise two-dimensional knot shape using a parametric formula.
It uses the height of the wiggles for the depth of the knot.
So if I tweak the height over here, it changes the height on the knot,
but it always looks correct from the top.
All right, that's it. Thanks for watching.