Some Simple Heegaard Splittings

Three 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 S^3.  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 S^3.

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 S^3.

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.

l23_traced

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 S^3, III is a genus one splitting of the mystery manifold M, 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.

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