# Unknown Witten-Reshetikhin-Turaev (WRT) Invariants for a 3 Manifold

I want to calculate WRT invariants for a project but all of the standard 3 manifolds seems to have been done already. I'm sort of new to this topic though so maybe I'm overlooking a particular example. Does anyone have any ideas for a 3-manifold, M, where WRT(M) hasn't been calculated yet?

EDIT: I should have specified closed and orientable 3-manifolds in the original question.

• I don't know anything about WRT, but what do you mean more precisely by "the standard 3-manifolds"? – user98602 Jul 25 '18 at 3:54
• @MikeMiller Good question, I should have specified closed and orientable 3-manifolds. To the best of my knowledge, that list includes, $S^3$, all the lens spaces including $RP^3$, homology spheres, Siefert 3-manifolds and $T^3$ – Bob Jul 25 '18 at 4:18
• What do you exactly mean by "unknown"? As far as I know WRT invariant (with level k) of a 3-manifold presented as a Dehn surgery along a link can be easily computed as a state sum of some colored Jones polynomial. – Henry Jul 26 '18 at 23:13

## 1 Answer

Seifert-fibered manifolds are the easy 3-manifolds! The JSJ decomposition + geometrization says that you can split every manifold along a small collection of tori into pieces which are either Seifert-fibered or hyperbolic. But the hyperbolic pieces are truly the place where 3-manifold theory gets crazy (and interesting): most 3-manifolds are hyperbolic!

One justification for this claim is as follows. First, knots/links in $S^3$ are split in a trichotomy: there are torus knots, there are satellite knots (which arise by "tracing out one knot around another", and so are in some sense reducible), and hyperbolic knots. Most knots (in an appropriate sense) are hyperbolic. A theorem of Thurston says that if you have a hyperbolic knot, Dehn surgery on $K$ can fail to be hyperbolic for only a finite number of slopes $p/q$; even better, it is conjectured that you can have at most 10 exceptional slopes. So a vast majority of the manifolds produced this way are hyperbolic!

So you could try doing computations on various hyperbolic manifolds, in particular, for various Dehn surgeries on hyperbolic knots/links. The Weeks manifold, the hyperbolic manifold with smallest volume, is a nice example (surgery on a relatively small link).

If you're doing this to get a better feel for WRT invariants, it seems like a good idea; if your goal is to do something novel enough to write a paper, I suggest you find a faculty mentor with similar interests to talk to about this; they will have much better suggestions than I do!

• Note that I have absolutely no idea what WRT invariants have been calculated or not. I am just going off of your comment above. – user98602 Jul 25 '18 at 4:33
• The Weeks manifold huh? I don't think I've come across that one. Not sure if anyone else has tried to find the WRT for that yet either. I'll get to work on that. Thanks! – Bob Jul 25 '18 at 5:30
• @Bob I warn that it may very well be that people have computed thousands of these things for hyperbolic manifolds, but I also think it might be a good project to think about. – user98602 Jul 25 '18 at 6:02