Which knot invariants have no known diagram-independent descriptions? Many knot invariants in knot theory are discovered by finding a property of knot diagrams which is invariant under the three Reidemeister moves.  Now in principle, any knot invariant can be described in a diagram-independent way, that is, as a property of the three-dimensional knot itself without reference to diagrams of the knot.  But in practice, it can take years between the development of a knot invariant and the discovery of a diagram-independent description of it.
So my question is, for what knot invariants is a diagram-independent description not yet known?
 A: I'd be happy to be proven wrong, but I would argue that this is still the case of the Jones polynomial and its generalizations, and my, maybe naive, understanding is that it's one of the many reasons it was considered fairly mysterious when it was discovered.
In fact, finding such a description was one of the motivations for Witten's Jones paper (he says as much in the introduction). As beautiful as this is, and even if like many others I'm more than happy to think about this as an actual definition, this is strictly speaking not mathematically rigorous and recovers only the values of the Jones polynomial at roots of unity.
Those can, I believe, now be given a diagrammatic-free definition in the framework of TFT's but this relies on fairly recent result. I also believe the case of generic $q$ is within reach but hasn't been done yet, and is closely related to exciting recent development in low-dimensional topology, e.g. finding a rigorous construction of analytic continuation of Chern-Simons theory, factorization algebras, the AJ conjecture etc..
