Reidemeister’s theorem

Despite living in 3-space our minds can only really grasp 2 dimensions since our eyes project the 3-dimensional world onto our  2-dimensional retinae. Nevertheless, we have a limited perception of 3 dimensions that comes from “layering” the different views of our retinae.

We perform a similar operation on tangles, projecting the inherently 3-dimensional objects onto a surface. Look at the example from the last post. It was necessary to draw it 2-dimensionally since computer displays are 2-dimensional. Nevertheless, we obtain a perception of the 3rd dimension by drawing “crossings” where one strand crosses over another. Try to locate the crossings in the example.

Example of a tangle

Example of a tangle

These projections are the most common way of representing tangles. They are called “tangle diagrams”. When we project a tangle diagram we take care to allow the only singularities (places where the projection doesn’t look nice) to be “transverse double points” which we represent as crossings. We don’t allow any of the following singularities: cusps, tangencies, or triple points.

Cusp

Cusp

Tangency

Tangency

Triple point

Triple point

We can guarantee that there are no such singularities by slightly tilting our projection if there are.

Now, how can we know if two tangle diagrams represent the same tangle? The answer is Reidemeister’s theorem: two tangle diagrams represent the same tangle if and only if they are connected by a sequence of Reidemeister moves. The pictures below demonstrate the 3 Reidemeister moves.

Reidemeister 1

Reidemeister 1

Reidemeister 2

Reidemeister 2

Reidemeister 3

Reidemeister 3

Looking at these pictures, it should be intuitively clear that performing Reidemeister moves does not change the tangle which a tangle diagram represents. The first Reidemeister move consists of adding or removing a “kink”. The second Reidemeister move consists of sliding strands past each other. The third Reidemeister move consists of moving a strand past a crossing. Look at the third move again and try to understand it physically: grab the middle strand and pull it through the crossing until it’s on the other side.

The difficult part of the theorem is proving the “only if” part, that is proving that the 3 Reidemeister moves suffice in order to transform one tangle diagram into any other tangle diagram which represents the same tangle. Notice, however that in the course of performing each of the Reidemeister moves we run afoul of our disallowed singularities. Perform a Reidemeister 1 on a physical tangle and as you get from one side of the equation to the other there will be a point in time where your projection is a cusp. Similarly, performing Reidemeister 2 will yield a tangency and Reidemeister 3 will yield a triple point.

One more important point is that Reidemeister moves are local. This means that if we have a large tangle we can perform Reidemeister moves on small pieces of the tangle. Let’s do an example to clarify.

Example of Reidemeister's theorem

Example of Reidemeister's theorem

We perform a single Reidemeister move locally in each equality. Try to identify where they occur.

Reidemeister’s theorem gives us the perfect tool for showing that two tangle diagrams represent the same tangle. Just perform Reidemeister moves to get from one diagram to the other. How can we show that two tangle diagrams represent different tangles? We may try to connect them via Reidemeister moves and fail, but that doesn’t show anything. Perhaps if we were smarter we could find the right sequence of Reidemeister moves. There’s an infinite number of such sequences so there’s no hope in testing them all. The answer to this puzzle is to look for invariants. But that’s the subject for another post!

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One Response to “Reidemeister’s theorem”

  1. Mep Says:

    Reidemeister 1 is EVIL!

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