  We introduce a family of 3-variable "Farey polynomials" that are closely
connected with the geometry and topology of $3$-manifolds and orbifolds as they
can be used to produce concrete realisations of the boundaries and local
coordinates for one-complex-dimensional deformation spaces of Kleinian groups.
As such, this family of polynomials has a number of quite remarkable
properties. We study these polynomials from an abstract combinatorial
viewpoint, including a recursive definition extending that which is known in
the literature for the special case of manifolds, even beyond what the geometry
predicts. We also present some intriguing examples and conjectures which we
would like to bring to the attention of researchers interested in algebraic
combinatorics and hypergeometric functions.
  The results in this paper additionally provide a practical approach to
various classification problems for rank-two subgroups of PSL(2,C) since they,
together with other recent work of the authors, make it possible to provide
certificates that certain groups are discrete and free, and effective ways to
identify relators.
