Last time, we used knot theory as a way of motivating these funny things called chord diagrams, which were circles with a collection of chords. They came up as ways of writing singular isotopy classes of singular knots, but there are many other ways of thinking of them. They can also be thought of as fixed-point free involutions on elements (modulo cyclic permutations) or a trivalent graph with a distinguished Hamiltonian cycle. I list these other incarnations more as a way of appealing to a broader base, though the latter perspective will be relevant today.

The topic of the day is the 4T relation, which I mentioned at the end of the previous post. It, together with the 1T relation, determined which functions on the set of all chord diagrams of a given degree came from knot invariants. The 1T relation came from a fairly straight-forward observation, while I didn’t even attempt to defend the 4T relation.

One way of thinking of this relation is as a symptom of a larger structure. We will generalize chord diagrams to a slightly broader context, and impose a relation there that is more natural. From there it is obvious to see that these generalized chord diagrams can always be reduced in terms of the usual chord diagrams, and that the only evidence of this broader structure is the 4T relation.