Math Circle Session 3: Knots

Today saw the third session of the primary school math circle we’ve been running for kids in Year 2 to Year 6. (I was absent for Session 2, where my wife covered mathematical card tricks.)

Today we looked at knots, drawing inspiration from Rozhkovskaya’s book as well as Adams’ introduction to knot theory. In particular, we covered the following points:

(Medial) graph representations of knot projections, moving backward and forward between them (see http://en.wikipedia.org/wiki/Knots_and_graphs)
The importance of assigning an (under/over) decision to crossings
Getting the kids to think of their own ideas for what knot equivalence might mean (shape, size, rotation, deformation – of what type) etc.

The key innovation we used here, which I think really brought the session to life, was to get the kids actually physically making the knots using Wikkistix, wax-coated string which allowed them to make, break, and remake their knots.

The kids really ran with this, and made their own discoveries, in particular:

One child discovered that some knots corresponding to the graph $K_3$ could be manipulated to produce the unknot, some could not.
A child discovered that it is possible to produce two interconnected knots, forming a link. Another child came to the same conclusion from the graph representation.
Consider the graph with vertex set $\{v_1,\ldots,v_n\}$ and edge set $\{\{v_i,v_{i+1}\} | 1 \leq i \leq n-1\}\} \cup \{\{v_n,v_1\}\}$ (is there a name for these graphs?). One child completely independently found that for $n$ even, this corresponds to a link of two knots, whereas for $n$ odd, it corresponds to a single knot.