Check out this exciting new preprint by Vaughan Jones!
V.F.R. Jones, Some Unitary Representations of Thompson’s Groups and , arXiv:1412.7740.
Links occur as braid closures, and so links can be studied via braid theory. This is the starting point for the Jones polynomial , and it’s very nice, because braids form a group (good algebraic property) which is orderable (even biorderable) and automatic (even biautomatic). But here are some complaints I have:
- Combing a link introduces an artificial structure- a `timeline’ where strands always `move forward through time’- which a link doesn’t naturally have. By trading links with braids, maybe we’re missing out on some essential je ne sais quoi that makes a link a link.
- More concretely, all known quantum link invariants come from Lie (bi)algebra constructions. But not all subfactors arise this way. For example, the Haagerup subfactor does not. If we believe that quantum invariants should correspond to finite index subfactors and not just to e.g. quantum groups, then surely braids are the wrong way to go.
- Everyone and their cousin uses braid groups nowadays.
Jones’s manuscript provides something new, that is a way to obtain links from elements of Thompson groups, and elements of Thompson groups from links. His algorithm seems just as natural as combing tangles to obtain braids. And it leads directly to new polynomial link invariants.
Thompson groups may be just as nice as braid groups. At the very least, they’re new in this context. It’s an open problem whether they are orderable, automatic, or amenable- as far as I know, they certainly might be. Indeed, they might be *much nicer*, because, unlike braid groups on more than two generators, they don’t contain a subgroup isomorphic to the free group on two generators.
As far as I know, Thompson groups have only shown up in mathematics so far as counterexamples. But in Jones’s paper, they naturally pop out of the structure of planar algebras. There’s nothing artificial about the correspondence between knots and Thompson group elements.
Edit: Yemon Choi points out a very nice interpretation of Thompson group elements which both sense in context and also in which the Thompson group is not a counterexample. The reference is HERE. What I said about Thompson groups only being counterexamples is even more false than that- they’ve featured before in quantum topology (and I wonder what the relation is, if any, with the construction in Jones’s new preprint).
Silly convention: Elements of braid groups are called braids. I suggest that elements of Thompson groups be called Thompsons. Indeed, two (dependent) elements of the Thompson group are used to construct links, and so, to be precise: What is at hand is truly a case of Thompson and Thompson.
Edit: Scott Carter points out that “Thompson and Thompson” ought to be Thomson and Thompson. Which makes sense to me- the link is a diagrammatic inner product of one Thompson with another Thompson acted on by a representation . Perhaps it’s reasonable to call a new polynomial invariant the `Thomson and Thompson Polynomial’ after all, with the `p’ reminding us of the action of .
Jones investigates the scaling limit of tangle diagrams, by letting the number of boundary points of the tangle fill out the boundary disc of the diagram (tangles with more and more strands). It wouldn’t make much sense, at least diagrammatically, to do this all at once- what would a tangle with uncountably many endpoints look like?- but what Jones does instead is to present a directed set construction, increasing the number of tangle endpoints inductively by gluing on tangles with . This mimics the block spin renormalization procedure from physics on a diagrammatic level. Jones credits Dylan Thurston with having suggested this idea to him.
Because the only way to add endpoints to a tangle is with caps or with cups, the number of intervals between tangle endpoints is always even. Jones chooses the interval endpoints (tangle endpoints) to correspond to dyadic rationals, that is to numbers of the form for some . The group of PL homeomorphisms of whose non-differential points are the dyadic rationals and whose slopes are all powers of is Thompson’s group , and this is how Thompson groups enter the picture.
The preprint opens up vast new vistas, and the relationship between links and Thompson groups which it points out is at once so powerful and so natural (e.g. the analogous construction for a different planar algebra gives the Tutte polynomial of a planar graph) that it would be difficult to imagine the new polynomials which his construction defines not once again reshaping major parts of quantum topology.
Two challenges are presented at the end of the preprint:
- Links are obtained from Thompsons by an inner product, that is by pairing a tangle obtained from the Thompson with a mirror image of another tangle obtained in a different way from the same Thompson. This sort-of reminds me of things like Heegaard splittings by the way, although of course that’s a very different idea. Anyway, the -index of a link is the smallest number of leaves of a Thompson (presented as a pair of rooted binary trees) required for this construction. Investigate bounds for the -index.
- Find and prove Markov Theorem for Thompson group presentations. This has got to be a major goal. I would be totally amazed if the `Markov moves’ for Thompsons were anything other than the set of moves on pairs of trees corresponding to the Reidemeister moves, as Jones himself suggests. This makes Thompsons *much more* natural group element representations of tangles than braids are.
Another obvious question is how to compute the polynomial invariant defined in this preprint, at least for small examples, and to establish whether it satisfies a reasonable-looking skein relation.
To complain (never forget to complain!), like the Jones polynomial, the construction of this invariant also uses planarity in an essential way, and therefore it is difficult to see how its construction might extend to virtual and to welded tangles.
I thank Ian Agol for drawing my attention to this preprint.