  The fixed-point index of a homeomorphism of Jordan curves measures the number
of fixed-points, with multiplicity, of the extension of the homeomorphism to
the full Jordan domains in question. The now-classical Circle Index Lemma says
that the fixed-point index of a positive-orientation-preserving homeomorphism
of round circles is always non-negative. We begin by proving a generalization
of this lemma, to accommodate Jordan curves bounding domains which do not
disconnect each other. We then apply this generalization to give a new proof of
Schramm's Incompatibility Theorem, which was used by Schramm to give the first
proof of the rigidity of circle packings filling the complex and hyperbolic
planes. As an example application, we include outlines of proofs of these
circle packing theorems.
  We then introduce a new tool, the so-called torus parametrization, for
working with fixed-point index, which allows some problems concerning this
quantity to be approached combinatorially. We apply torus parametrization to
give the first purely topological proof of the following lemma: given two
positively oriented Jordan curves, one may essentially prescribe the images of
three points of one of the curves in the other, and obtain an
orientation-preserving homeomorphism between the curves, having non-negative
fixed-point index, which respects this prescription. This lemma is essential to
our proof of the Incompatibility Theorem.
