My mum has always thought that she's not mathematical; actually always been told by
her school, by various different people that she's not mathematical but there are always a few clues
that I saw throughout my life, I thought, hang on I actually think you might be good at maths. So
first of all she's pretty good at textiles - my mum once crocheted me this wonderful festive poncho,
the boundary of which is a digon; and it doesn't lie flat so you're crocheting in hyperbolic space.
And also on all of her letters, any like birthday cards, everything I get, she always draws
this little symbol. - Everything, it's my seal. - So this little symbol, you'll recognize this as the
3_1 knot - it's got three crossings, it is a closed knot, and it is the smallest non-trivial
prime knot. To seal in mum's mathematical abilities and embrace her into the world
of maths I thought I would give her a mathematical embrace. So welcome to: Ayliean's guide on how to give
a mathematically interesting hug...featuring Lorna. So let's investigate first of all normal- normal
hugs. So there's the first one here, if you're just holding hands, this is not a hug. This would be like
the unknot, it's just a loop not a hug at all. If you were to look at a normal hug, so usually
one person goes over, one person goes under - I am the shorty of the family so I always go under -
this would be your normal hug, which is two linked loops. And I don't think it's great because,
without unlinking, you can just- you can escape like that. So it's two- it's two unhugs joined together.
Every so often you get a- kind of the awkward hug where you don't know if someone's gonna go
over or under, and you end up doing one person over, one person under, which looks like this.
Which is better because you are linked - but we can do- we can take that one step further, we
can do even better than that. Here is your symbol in a hug, the trefoil knot, 3_1, as a hug. So
step number one, the person behind you're going to have to do the over under. So I'm going to go
left arm under, right arm over. Then, you're going to reverse that situation - so cross
over here and you just need to hold my hands. Cool. And this, although it doesn't look like it,
is a trefoil hug. We can do a continuous deformation; so we do it like this,
there is the trefoil knot, that's the trefoil hug. I'm sure you can like draw on there to show that
it is definitely a trefoil hug. We've got three crossings and we're completely linked, we're all
tied together. There is something I do want to show you about the chirality of that hug though.
So the way that mum draws this, it just kind of loops all together but you can draw this as an
alternating knot. All knots which have less than eight crossings are alternating. So if I start here-
so here it is as an alternating knot
where it goes under, over, under, over, under - so it continues with the under, over. But there is another.
So you can do this again but instead of starting going over
you can start going under. So if I draw the exact same knot but with an under to start with -
this is like doing something left handed for me -
hopefully I haven't drawn the same knot - nope, it's all good. Okay, so these two knots, they
both have three crossings, they're both a trefoil knot but they are different, they have chirality,
which is like a handedness. So they've got a left-handed knot and a right-handed knot.
Chirality, such an interesting symmetry, and you can see that in the hug. So back to-
back to the hugs. So if we did the one that we did before, where we do one over, one under, cross
it, join - you can only deform this in one way. So you can only do it like this
where it goes over the left hand side. You want to go back in? If you try and go the other way
it- it doesn't work, it doesn't work because of the chirality, it only has one direction. So if
you wanted to do the other sided knot, rather than going left hand under, right hand over
I would have to go this way round and this one's over, so it's going to go under,
join - and then now we should be able to go the other way.
So, chirality! - (Brady: Which hug did you like best, mum?) - Umm just the plain old good
hug, yeah. - (She's over complicating things) - It's always okay to hug your mum, any way you like.
I have tried - this is gonna make me sound like such a sad person -
I've tried to do these hug knots on my own. It doesn't work. - (How- what do you even do?) - Well so this
this trefoil knot, the 3_1 knot, is sometimes called the overhand knot?