College Math Teaching

July 1, 2015

Embarrassing gaps in my mathematical knowledge

Filed under: mathematician, topology — Tags: , — collegemathteaching @ 1:56 pm

Yes, mathematics is a huge, huge subject and no one knows everything. And, when I was a graduate student, I could only focus on 1-2 advanced courses at a time, and when I was working on my thesis, I almost had a “blinders on” approach to finishing that thing up. I think that I had to do that, given my intellectual limitations.

So, even in “my area”, my knowledge outside of a very narrow area was weak at best.

Add to this: 20+ years of teaching 3 courses per semester; I’ve even forgotten some of what I once knew well, though in return, I’ve picked up elementary knowledge in disciplines that I didn’t know before.

But, I have many gaps in my own “area”. One of these is in the area of hyperbolic geometry and the geometry of knot complements (think of this way: take a smooth simple closed curve in R^3 , add a point at infinity to get S^3 (a compact space), now take a solid torus product neighborhood of the knot (“thicken” the knot up into a sort of “rope”) then remove this “rope” from S^3 . What you have left over is a “knot complement” manifold.

Now these knot complements fall into one of 3 different types: they are torus knot complements (the knot can live on the “skin” of a torus),

torusknot

satellite knot complements (the knot can live inside the solid torus that is the product neighborhood of a different, mathematically inequivalent knot,

satelliteknot

or the knot complement is “hyperbolic”; it can be given a hyperbolic structure. At least for “most” knots of small “crossing number” (roughly: how many crossings the knot diagram has), are hyperbolic knots.

So it turns out that the complement of such knots can be filled with “horoballs”; roughly speaking, these are the interior of spheres which are “tangent to infinity”; infinity is the “missing stuff” that was removed when the knot was removed from S^3. And, I really never understood what was going on at all.

horo_fig8

I suppose that one can view the boundary of these balls (called “horospheres”) as one would view, say, the level planes z = k in R^3 ; those planes become spheres when the point at infinity is added. This is a horoball packing of the complement of the figure 8 knot; missing is the horosphere at z = 1 which can be thought of as a plane.

But the internet is a wonderful thing, and I found a lecture based on the work of Anastasiia Tsvietkova and Morwen Thistlethwaite (who generated the horoball packing photo above) and I’ll be trying to wrap my head around this.

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