  The idea that the elementary particles might have the symmetry of knots has
had a long history. In any current formulation of this idea, however, the knot
must be quantized. The present review is a summary of a small set of papers
that began as an attempt to correlate the properties of quantized knots with
the empirical properties of the elementary particles. As the ideas behind these
papers have developed over a number of years the model has evolved, and this
review is intended to present the model in its current form. The original
picture of an elementary fermion as a solitonic knot of field, described by the
trefoil representation of SUq(2), has expanded into its current form in which a
knotted field is complementary to a composite structure composed of three or
more preons that in turn are described by the fundamental representation of
SLq(2). These complementary descriptions may be interpreted as describing
single composite particles composed of three or more preons bound by a knotted
field.
