Posts Tagged ‘math.GN’

Chord Diagrams: Understanding the 4T Relation

December 18, 2007

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 $2*degree$ 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.

Chord Diagrams

December 13, 2007

Well, its been quite awhile since the last post.  The real problem is that this blog can’t be a higher priority for me than, you know, important stuff like learning, teaching and research.  So, when time and energy get tight, its one of the first things to go.  It also doesn’t help that the more I bend my mind towards research, the less time I spend thinking about things that would actually make good posts (I have started and abandoned several posts on uninteresting research-type things in the last couple months).

I’ll try to get back into the swing of things by talking about some stuff I really enjoy, but is far from my research: knots.  I’ve always had a soft spot for knot theory, since its like a poor man’s number theory; a source of simple problems which require techniques from advanced math to solve.  A good example of this are Tait’s conjectures, three basic conjectures from the 19th century that resisted proof until the discovery of the Jones invariant using techniques from analysis and representation theory (and secretly physics).

Today, though, I’d like to talk about ‘chord diagrams’, a type of object subtly related to that of knots, and whose study can yield some interesting new knot invariants.  They also come up in a number of different areas (solving Feynman diagrams, the representation theory of lie algebras) that I know very little about.  If anyone reading is more familiar with some of the places these come up, please send me a reference.