To understand a knot K, we want to understand how it behaves, what we can (or can’t) do to it.
Since the knot has to stay intact, and since we’re on the outside, it’s better to look at the knot’s complement. That should be okay, because it has exactly the same shape as the thing itself, right? The Gordon-Luecke theorem is there to tell us that you can’t weirdly wrap the complement in ways not possible with the knot itself.
And besides, a knot’s inside has no room to maneuver loops in. So instead, we play lasso in the space left after we have carved out our knot!
Our test lasso shouldn’t tangle up with itself, because we already have a knot to study and wouldn’t want to complicate things. So you should think of it as being made of that mythical topological material, that can stretch indefinitely and pass through itself (but not the knot), in first approximation: a rubber band.
Then a lasso configuration A is different from another one B if we cannot deform A and B to match up positionally. For example, if A doesn’t wrap around K at all, and B does so once, they’re different!
Finding out the set G of all essentially different ways to tie a string around our knot K will tell us something fundamental about K. There’ll be an element for each number of times we can wrap our test lace around K. If that were all, our story would end here, but interestingly, sometimes there are more elements, sometimes less!
Before we get to these let me show you that the set G is actually a group. First we need an operation between lasso paths, a kind of addition to combine two of them into another one:
To add two paths A and B, first pick a base point P, then deform A and B to touch P. Cut both of them up there, and stick them together the other way. If you draw arrows on your paths, giving them an orientation, you’ll notice that only one of the ways to reglue the ends into a single loop makes their orientations agree. That’s the one we want!
Putting arrows on your paths is generally a good idea, because now we can have both, left and right windings around a piece of knot: all our elements have inverses, there’s a B for every (signed) integer. And A is a zero-element, adding nothing to any path!
So, there’s an operation: check, inverses: check, neutral element: check – we have a group! That’s the fundamental group!
It’s also called homotopy group because our rubber lasso material is what your innocent-looking topologist neighbour likes to tie together his proofs with, that two particular manifolds he brought with him last night, are indeed homotopy equivalent.
So, it is a knot invariant, you can use it to prove that two knots are not equivalent under isotopy (in R³). knot group isn’t a complete invariant, so sometimes it cannot distinguish different knots. There are many more knot invariants:
Example: Picture a line’s complement (in R³). It is homotopic (in R³-L) to a circle, and its homotopy group is Z: the integers with addition. R³ without two lines gives the free group over two generators. Every group is a subgroup of a free group and also a quotient of one.
So we can expect that any knot’s complement’s fundamental group can be written as a quotient of a free group over some equalities that describe the new freedom to move the lasso we get when linking line segments among each other instead with the infinite far away. Here’s the one for the trefoil knot:
also known as Artin’s braid group B_3.
If you find that unconvincing, maybe this page is for you:
It’s where I stumbled over this post, whereentangles the knot groups with other fancy stuff:
Finally a word of warning: only the abelianization of the first homotopy group is the first homology group… Also, Milnor’s link group here is a bit but not quite similar, be careful:
I’m posting this as a warm-up for some more topological things I have in my queue. Stay tuned!