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.