Knot Exteriors

Topologists often study knots by looking at what remains when the knot is removed from .  I suppose they’re looking (somewhat ironically) at everything but the knot.   Hmm.  In any case, it works, so why don’t we let be a knot in and define the knot exterior to be , the closure of minus a regular neighborhood of the knot.

One way to visualize knot exteriors is to take the projection of to where the point at infinity is chosen to lie inside .  In this case, the knot exterior looks like a 3–ball minus a knotted arc inside.  Two such exteriors are drawn at the right.  I’ve added the knot (a neighborhood of which we removed, remember) in blue.  The first is the exterior of the unknot, and the second is the exterior of a trefoil knot.    It’s pretty easy to see in the first case that the exterior is a solid torus.

The converse of this fact, that if is homeomorphic to a solid torus then is the unknot, is also true, because a compressing disk for the solid torus gives a Seifert surface for the knot.

Some Simple Heegaard Splittings

Consider picture I, in which we have two balls with blue circles on their boundary.  It’s not too hard to see that if we glue these two balls together so that the blue curves on their boundaries are identified, we obtain the space .  In picture II we have two solid tori with blue curves, and again, if we identify the boundaries of these two in such a way that the blue curves are identified, we obtain .

This second one might be slightly harder to see, so let’s look at it another way: take the first solid torus, and glue a thickened disk along its edge to the blue curve.   We consider the thickened disk to be a neighborhood of the disk bounding the blue curve in the second solid torus.  What’s left after we’ve glued this disk to the first torus?  A 3-ball.  Glue that sucker in, you get .

In picture III we ‘ve changed things up a bit.  We still have to solid tori glued together, but this time the blue curves are different.  It’s an exercise for the reader to determine this space.  As a hint, it might be easier to see if you cut both solid tori along the disks bounding the blue curves.

An even more complicated (and difficult to visualize) example is pictured here.  In the top line I’ve written it as a union of solid tori, and the second line it’s written as a thickened disk attached to the trefoil knot in the boundary of the second solid torus.

These are all examples of decompositions of 3-manifolds called Heegaard splittings.  The first two are genus zero splittings of , III is a genus one splitting of the mystery manifold , and the last is a genus one splitting of a thingy called a “lens space.”

The point of all this is to see how a knot in bridge position gives a Heegaard splitting for a manifold associated to the knot, which I will write about next time.

Bridge Number

Consider the pictured trefoil knot embedded in .  The plane is shown, and the intersections of the plane with are marked in blue.  The knot is drawn with a dashed line where it lies below the plane; notice that the intersection of the lower half space with consists of three unknotted arcs.  Similarly, the intersection of the upper half space with is three unknotted arcs.  These arcs look a bit like bridges, and we say this is a three bridge presentation of .

The smallest integer such that a knot has an bridge presentation is called the bridge number of .  The above picture shows that the bridge number of is less than or equal to three.  We can do better though!  Below is a two bridge presentation of the trefoil.  Since a bridge number one knot is the unknot, this shows that the trefoil has bridge number two.

An incompressible surface

Have a look at the this torus, embedded in a standard way in .  The shaded disk is inside the torus with its boundary on the torus itself.  Consider , the boundary of .  It represents a nontrivial loop in the torus, but since it bounds a disk in , the loop is trivial in .  Such a disk is called a compressing disk for the torus, and we can compress the torus by cutting along and gluing in two copies of as shown in the second picture.  This is also called surgering along .  In this case, we obtain a sphere by compressing along .

Notice that when we compress along we decrease the genus of our surface by one.  With that in mind, consider the Seifert surface for the knot we looked at previously.  Are there any compressing disks for ?  Suppose there was a compressing disk and didn’t separate into two components .  Then we could compress along , obtaining a new surface .  Since we didn’t touch anything near the boundary of , is still a Seifert surface for .  But is a punctured torus, so must be a disk, and the only knot bounding a disk is the unknot.

On the other hand, if there was a compressing disk such that did separate into two components, then one of these components would have to be an annulus.   As above, we would get a disk bounding the knot .  (Why?  Working out these details is a fun exercise using Euler characteristic arguments.)

So if you believe that the trefoil is knotted, then the surface must be incompressible!

Seifert Surfaces

Hello!  I’m Sean, a graduate student here at the University of Texas and an aspiring topologist.  Topologists seem to like pictures as a general rule, and I’m no exception, so I hope to post some nice pictures of interesting mathematical objects.

Speaking of, the picture on the right is a trefoil knot.  (Isn’t it pretty?)  We usually think of such knots as sitting inside (or if you like).

Now it’s a fact that given any knot , there’s an embedded orientable surface in whose boundary is .  Such a surface is called a Seifert surface or a spanning surface.  The picture on the right shows a Seifert surface for the trefoil.  As you can see, it consists of two regions on the left and right joined by three twisted bands.  We can tell that it’s orientable because there’s no way to cross from the “top” to the “bottom” of the surface.  In other words, there are no embedded Mobius bands in the surface.

What kind of surface is this?  We know it has one boundary component and that it’s orientable, so from the classification of orientable surfaces we know that this must be a surface of genus with one puncture.  To find out the genus of the surface, we can cut along the top two twists, being careful to keep track of how we cut.  Then we can untwist at the bottom and straighten the surface out so that it lies flat.  Finally, we can glue back the pieces marked with arrows as indicated.  Hurray!  Now we recognize the surface as a punctured torus!

(Cognoscenti will note that we could have seen immediately  that a once punctured orientable surface with the homotopy type of the wedge of two circles must be a punctured torus.  I think it’s more fun to draw pictures.)