  These notes summarize the lectures delivered in the V Mexican School of
Particle Physics, at the University of Guanajuato. We give a survey of the
application of Ashtekar's variables to the quantization of General Relativity
in four dimensions with special emphasis on the application of techniques of
analytic knot theory to the loop representation. We discuss the role that the
Jones Polynomial plays as a generator of nondegenerate quantum states of the
gravitational field.
