The analogy doesn't quite give a number theoretic version of the Poincare conjecture.  See Sikora, "Analogies between group actions on 3-manifolds and number fields"
(arXiv:0107210): the author states the Poincare conjecture as "S<sup>3</sup> is the only closed 3-manifold with no unbranched covers."  The analogous statement in number theory is that Q is the only number field with no unramified extensions, and indeed he points out that there are a few known counterexamples, such as the imaginary quadratic fields with class number 1.

The paper also has a nice but short summary of the so-called "MKR dictionary" relating 3-manifolds to number fields in section 2.  Morishita's expository article on the subject, arXiv:0904.3399, has more to say about what knot complements, meridians and longitudes, knot groups, etc. are, but I don't think there's an explanation of what knot surgery would be and so I'm not sure how Kirby calculus fits into the picture.

Edit: An <a href="http://arxiv.org/abs/math/0602064" title="arxiv">article</a> by B. Morin on Sikora's dictionary (and how it relates to Lichtenbaum's cohomology, p. 28): "he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora's formulas which leads us to clarify and to extend the dictionary of arithmetic topology."