Does the Jones Polynomial Detect the Unknot ?

@inproceedings{Bigelow2002DoesTJ,
  title={Does the Jones Polynomial Detect the Unknot ?},
  author={Stephen Bigelow},
  year={2002}
}
We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. I will describe a computer algorithm to search for such a braid.