  In their paper `A new algorithm for recognizing the unknot', in Geometry and
Topology', 2 (1998) n. 9, 175-220, the first author and Michael Hirsch
presented a then new algorithm for recognizing the unknot. The first part of
the algorithm required the systematic enumeration of all discs which support a
`braid foliation' and are embeddable in 3-space. The boundaries of these
`foliated embeddable discs' (FED's) are the collection of all closed braid
representatives of the unknot, up to conjugacy, and the second part of the
algorithm produces a word in the generators of the braid group which represents
the boundary of the previously listed FED's. The third part tests whether a
given closed braid is conjugate to the boundary of a FED on the list. In this
paper we describe implementations of the first and second parts of the
algorithm. We also give some of the data which we obtained. The data suggest
that FED's have unexplored and interesting structure. Open questions are
interspersed throughout the manuscript. The third part of the algorithm was
studied by the first author, H. K. Ko and S. J. Lee in their paper `A new
approach to the word and conjugacy problems in the braid groups', in `Advances
in Mathematics', 139 (1998), 322-353, and in their preprint `The infimum,
supremum and geodesic length of a braid conjugacy class',
arXiv:math.GT/0003125, and implemented by S. J. Lee in his preprint
`Implementation of an algorithm for solving the conjugacy problem in the braid
groups' (sangjin@knot.kaist.ac.kr). At this writing his algorithm is polynomial
for n less than or equal to 4 and exponential for n greater than or equal to 5.
