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A theorem of Kirby gives a necessary and sufficient condition for two framed links in    S  3    to yield orientation‐preserving diffeomorphic results of surgery. Kirby's theorem is an important method for constructing invariants of    3   ‐manifolds. In this paper, we prove a variant of Kirby's theorem for null‐homologous framed links in a    3   ‐manifold. This result involves a new kind of moves, called IHX‐moves, which are closely related to the IHX relation in the theory of finite type invariants. When the first homology group of   M   is free abelian, we give a refinement of this result to    ±  1   ‐framed, algebraically split, null‐homologous framed links in   M  .

Kirby calculus for null‐homologous framed links in 3 ‐manifolds