My two high-school kids came by today. We’ve been trying to get a new teaching session together since early February, but they had a hell of a time all through February, and all our appointments ended up canceled with little or no notice; and then I spent March and April on tour.
We pressed on with knot theory. Today, we discussed knot sums, prime knots, knot tabulation, behavior of the one invariant (n-colorability) we know so far under knot sums, Dowker codes, and we got started on Conway codes for knots. Next week, I plan for us to finish up talking about the Conway knot notation, get the connection between rational knots and continued fractions down pat, and start looking into new invariants.
- February 1st, 2008
- 2:27 pm
So, here’s the plan for my 10th grade topology students.
Today, we’ll abandon algebraic topology completely, and instead go into knot theory. I’ll want to discuss what we mean by a knot (embedding of in ), what we mean by a knot deformation (thus introducing isotopies while we’re at it) and the Reidemeister moves. Also we’ll discuss knot invariants – and their use analogous to topological invariants.
Later on, we’ll continue with other invariants; definitely including the Jones polynomial, and possibly even covering Khovanov homology. One possible end report would be to explain a bunch of knot invariants and show using examples how these have different coarseness.
Edited to add: I got myself some damn smart students. They figured out the Reidemeister moves on their own – as well as minimal crossing number in a projection being highly relevant – with basically no prompting from me. I’m impressed.
- December 14th, 2007
- 4:31 pm
The last meeting with my 10th grade topology kids this year just finished. We introduced singular homology, calculated the singular homology of a point and discussed homeomorphism invariance.
Next term, we’ll want to show homotopy invariance and that the singular and simplicial homology coincide when applicable. After that, we’ll change directions slightly.
The future after that holds knot theory, was decided today. We’ll want to introduce knots, look at Reidemeister moves and basic knot invariants. I use basic here in a pretty wide sense – we’ll probably do the Jones polynomial and we might even end up doing Khovanov homology if I feel particularly insane late spring.
- November 16th, 2007
- 4:34 pm
Today, I told my two bright students about abstract and geometric simplicial complexes, about the boundary map and the chain complex over a ring R associated with a simplicial complex Δ, and assigned them reading out of Hatcher’s Algebraic Topology.
The next couple of weeks will be spent doing homology of simplicial complexes, singular homology, equivalence of the two, neat things you can do with them; and then we’ll start moving towards a Borsuk-Ulam-y topological combinatorics direction.
I might end up pulling combinatorics papers from my old “gang” in Stockholm on graph complexes, and graph property complexes, and poke around those with them.