My, what a tangled web we weave

Hi, my name is Eitan. I’m a student of mathematics. Welcome to my personal blog. I intend to post mathematical exposition, but since this is a personal blog I will also post political thoughts or anything else that comes to mind. Let’s start with some math.

What’s a tangle? Tangles are very important objects in topology. Physically, they are just a number of strings in our usual 3-dimensional space whose endpoints (if they have endpoints) are attached to some boundary surface. We allow the strings to move around freely in 3-space so long as the endpoints remain fixed and we cannot pass strings through each other (or over endpoints). We also allow the strings to stretch and compress as though they’re made of rubber. Here’s an example of a tangle:

Example of a tangle

Example of a tangle

Try to count how many strings there are.

Tangles are really generalizations of knots and links. A knot is a tangle made of only 1 closed string, meaning the string has no endpoints, it’s just a circle. A link is a tangle made of any number of closed strings. Notice that all knots are links and that all links are tangles. Another way of thinking about tangles is that they are “local” pictures of knots, that is, if we zoom in on a knot and look at a small neighborhood, that neighborhood will contain a tangle.

Let’s look at examples of knots and links.

Trefoil Knot

Trefoil Knot

Hopf Link

Hopf Link

By tracing along with your finger, verify that the trefoil knot has only 1 string while the Hopf link has 2 strings.

Recall that I said the endpoints of the strings in a tangle must be attached to some boundary surface. In the example the boundary surface comes in two pieces, a bottom and a top. This is one convention for tangles, the “monoidal category” convention. Another convention, the “planar algebra” convention, is that the surface has only one piece. Really, there’s no important difference between thinking in either convention. It’s only a question of convenience for a given application.

I think that’s enough for now. Stay tuned to hear about Reidemeister’s theorem.

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5 Responses to “My, what a tangled web we weave”

  1. Akhil Mathew Says:

    So is a tangle rigorously a “homotopy class of 1-submanifolds of R^3” or something like that?

  2. Eitan Says:

    Woohoo! The first real comment on my blog! A tangle is rigorously something like that yes. Instead of homotopy though, the actual relation is called ambient isotopy. An ambient isotopy between links K_0,K_1 is a 1-parameter family of diffeomorphisms F_t of the ambient space \mathbb{R}^3 such that F_0 is the identity and F_1(K_0)=K_1. In the case of tangles, we require that the ambient isotopy fixes the boundary of our ambient space (and hence the boundaries of our tangles).

  3. Akhil Mathew Says:

    Thanks for the explanation!

    So what happens then if we drop the assumption that the object is one-dimensional, and that it is a subset of R^3 (though perhaps it needs to be compact or something)? Can we consider higher-dimensional analogs of knots in some other manifold, up to an ambient isotopy?

  4. Eitan Says:

    Indeed we can and do consider such things. In general we may consider n-manifolds embedded in n+k dimensions and there is a conjecture about what sort of algebraic structure describes the set of all such things called the tangle hypothesis.

    One important application of this is to categorify link invariants. As with algebraic topology, one can turn numerical invariants like Euler characteristic into functorial invariants like simplicial homology. Then functoriality should mean that a surface in 4 dimensions should induce maps between the homologies of links which form the boundary of the surface. Hopefully, I will post more about this stuff at some point.

  5. Akhil Mathew Says:

    Interesting, thanks!

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