Restarting high school topology

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.

Anyone have a favorite invariant that you'd like me to talk about? I'm hoping (in my wildest most bizarre dreams) to get around to the Alexander polynomial and possibly even talk about Khovanov homology, but that depends a LOT on whether they're prepared to continue through their summer holidays or not - and even then I doubt we'll make it up to Khovanov.

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