The SLq(2) extension of the standard model

The idea that the elementary particles might have the symmetry of knots has had a long history. In any modern 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 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 present form in which a knotted field is complementary to a composite structure composed of three preons that in turn are described by the fundamental representation of SLq(2). Higher representations of SLq(2) are interpreted as describing composite particles composed of three or more preons bound by a knotted field. This preon model unexpectedly agrees in important detail with the Harari-Shupe model. There is an associated Lagrangian dynamics capable in principle of describing the interactions and masses of the particles generated by the model.