The End 2016 Mathematics A To Z: Unlink

This is going to be a fun one. It lets me get into knot theory again.

Unlink.

An unlink is what knot theorists call that heap of loose rubber bands in that one drawer compartment.

The longer way around. It starts with knots. I love knots. If I were stronger on abstract reasoning and weaker on computation I’d have been a knot theorist. At least graph theory anyway. The mathematical idea of a knot is inspired by a string tied together. In making it a mathematical idea we perfect the string. It becomes as thin as a line, though it curves as much as we want. It can stretch out or squash down as much as we want. It slides frictionlessly against itself. Gravity doesn’t make it drop any. This removes the hassles of real-world objects from it. It also means actual strings or yarns or whatever can’t be knots anymore. Only something that’s a loop which closes back on itself can be a knot. The knot you might make in a shoelace, to use an example, could be undone by pushing the tip back through the ‘knot’. Since our mathematical string is frictionless we can do that, effortlessly. We’re left with nothing.

But you can create a pretty good approximation to a mathematical knot if you have some kind of cable that can be connected to its own end. Loop the thing around as you like, connect end to end, and you’ve got it. I recommend the glow sticks sold for people to take to parties or raves or the like. They’re fun. If you tie it up so that the string (rope, glow stick, whatever) can’t spread out into a simple O shape no matter how you shake it up (short of breaking the cable) then you have a knot. There are many of them. Trefoil knots are probably the easiest to get, but if you’re short on inspiration try looking at Celtic knot patterns.

But if the string can be shaken out until it’s a simple O shape, the sort of thing you can place flat on a table, then you have an unknot. Just from the vocabulary this you see why I like the subject so. Since this hasn’t quite got silly enough, let me assure you that an unknot is itself a kind of knot; we call it the trivial knot. It’s the knot that’s too simple to be a knot. I’m sure you were worried about that. I only hear people call it an unknot, but maybe there are heritages that prefer “trivial knot”.

So that’s knots. What happens if you have more than one thing, though? What if you have a couple of string-loops? Several cables. We know these things can happen in the real world, since we’ve looked behind the TV set or the wireless router and we know there’s somehow more cables there than there are even things to connect.

Even mathematicians wouldn’t want to ignore something that caught up with real world implications. And we don’t. We get to them after we’re pretty comfortable working with knots. Describing them, working out the theoretical tools we’d use to un-knot a proper knot (spoiler: we cut things), coming up with polynomials that describe them, that sort of thing. When we’re ready for a new trick there we consider what happens if we have several knots. We call this bundle of knots a “link”. Well, what would you call it?

A link is a collection of knots. By talking about a link we expect that at least some of the knots are going to loop around each other. This covers a lot of possibilities. We could picture one of those construction-paper chains, made of intertwined loops, that are good for elementary school craft projects to be a link. We can picture a keychain with a bunch of keys dangling from it to be a link. (Imagine each key is a knot, just made of a very fat, metal “string”. C’mon, you can give me that.) The mass of cables hiding behind the TV stand is not properly a link, since it’s not properly made out of knots. But if you can imagine taking the ends of each of those wires and looping them back to the origins, then the somehow vaster mess you get from that would be a link again.

And then we come to an “unlink”. This has two pieces. The first is that it’s a collection of knots, yes, but knots that don’t interlink. We can pull them apart without any of them tugging the others along. The second piece is that each of the knots is itself an unknot. Trivial knots. Whichever you like to call them.

The “unlink” also gets called the “trivial link”, since it’s as boring a link as you can imagine. Manifested in the real world, well, an unbroken rubber band is a pretty good unknot. And a pile of unbroken rubber bands will therefore be an unlink.

If you get into knot theory you end up trying to prove stuff about complicated knots, and complicated links. Often these are easiest to prove by chopping up the knot or the link into something simpler. Maybe you chop those smaller pieces up again. And you can’t get simpler than an unlink. If you can prove whatever you want to show for that then you’ve got a step done toward proving your whole actually interesting thing. This is why we see unknots and unlinks enough to give them names and attention.

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