Remarks on refined Kirby calculus for three-manifolds of cyclic first homology groups of odd prime power orders

Abstract

Habiro arranged Kirby moves into a pair so that it preserves linking matrices. The author showed that two framed links of diagonal linking matrices yield homeomorphic 3-manifolds of linking form (±1/p) for an odd prime p if and only if they are related by a sequence of Habiro moves. We generalize this result to 3-manifolds of linking forms (±1/c) for any odd prime power c.