Combinatorial excursions in moduli space

Given an abstract triangulation of a torus, there is a unique point in moduli space which supports a circle packing for this triangulation. We will describe combinatorial deformations analogous to the process of conformal welding. These combi-natorial deformations allow us to travel in moduli space from any packable torus to a point arbitrarily close to any other torus we choose. We also provide two proofs of Toki’s result that any torus can be transformed into any other by a confor-mal welding and compute the maps necessary to accomplish the welding. 1. Introduction. A circle packing is a configuration of circles with a prescribed pattern of tangencies. The interplay between the combinatorial “pattern ” and the ge-ometry provided by the circles has generated intense interest in recent years. As a result, a discrete version of complex analysis based on maps betwee