The Local-Global Conjecture is False for Generalized Circle Packings

TL;DR

This paper extends the disproof of the Local-Global Conjecture from Apollonian packings to four other types of integral circle packings, revealing quadratic invariants and reciprocity obstructions affecting the curvatures.

Contribution

It introduces new quadratic invariants and reciprocity obstructions for multiple circle packings, broadening the understanding of when the conjecture fails.

Findings

Quadratic invariants imply reciprocity obstructions in certain packings

Explicit parametrization of tangent circles aids in analysis

Partial obstructions limit the set of possible curvatures

Abstract

Haag, Kertzer, Rickards, and Stange disprove the Local-Global Conjecture for Apollonian circle packings. We extend their disproof to four more types of integral circle packing: the octahedral, cubic, square, and triangular packings. In each case, we find quadratic invariants which imply quadratic reciprocity obstructions to the conjecture in certain packings. We utilize an explicit parametrization of circles tangent to a fixed circle in each packing type, and a quadratic reciprocity argument. Even in the packings where we do not find quadratic obstructions, the curvatures exhibit a predictable reciprocity structure. This leads to partial obstructions on integers appearing as curvatures in subsets of the packing.