The fixed-point index of a homeomorphism of Jordan curves measures the numberof fixed-points, with multiplicity, of the extension of the homeomorphism tothe full Jordan domains in question. The now-classical Circle Index Lemma saysthat the fixed-point index of a positive-orientation-preserving homeomorphismof round circles is always non-negative. We begin by proving a generalizationof this lemma, to accommodate Jordan curves bounding domains which do notdisconnect each other. We then apply this generalization to give a new proof ofSchramm's Incompatibility Theorem, which was used by Schramm to give the firstproof of the rigidity of circle packings filling the complex and hyperbolicplanes. As an example application, we include outlines of proofs of thesecircle packing theorems. We then introduce a new tool, the so-called torus parametrization, forworking with fixed-point index, which allows some problems concerning thisquantity to be approached combinatorially. We apply torus parametrization togive the first purely topological proof of the following lemma: given twopositively oriented Jordan curves, one may essentially prescribe the images ofthree points of one of the curves in the other, and obtain anorientation-preserving homeomorphism between the curves, having non-negativefixed-point index, which respects this prescription. This lemma is essential toour proof of the Incompatibility Theorem.
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