  In the loop representation the quantum constraints of gravity can be solved.
This fact allowed significant progress in the understanding of the space of
states of the theory. The analysis of the constraints over loop dependent
wavefunctions has been traditionally based upon geometric (in contrast to
analytic) properties of the loops. The reason for this preferred way is
twofold: for one hand the inherent difficulties associated with the analytic
loop calculus, and on the other our limited knowledge about the analytic
properties of knots invariants. Extended loops provide a way to overcome the
difficulties at both levels. For one hand, a systematic method to construct
analytic expressions of diffeomorphism invariants (the extended knots) in terms
of the Chern-Simons propagators can be developed. Extended knots are simply
related to ordinary knots (at least formally). The analytic expressions of knot
invariants could be produced then in a generic way. On the other hand, the
evaluation of the Hamiltonian over extended loop wavefunctions can be
thoroughly accomplished in the extended loop framework. These two ingredients
promote extended loops as a potential resort for answering important questions
about quantum gravity.
