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I am wondering if there are links $L_1, L_2$ in the sphere $S^3$ such that:

  1. the signatures of $L_1, L_2$ are known.
  2. we do not know if they are related by Kirby moves.

If so, could you specify the links? Or if it is known that there is a method to check if links are related by Kirby moves or not, please let me know.

Thank you.

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    $\begingroup$ What does it mean to say that links are related by Kirby moves? The phrase usually refers to framed links, where Kirby moves preserve the diffeomorphism type of the manifold obtained by surgery with the given framings. $\endgroup$ – Danny Ruberman Aug 28 '13 at 2:41
  • $\begingroup$ @DannyRuberman Yes, these links should be framed links. Thanks. $\endgroup$ – Primo Aug 28 '13 at 2:56
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The question is not well-posed as it stands, so I will attempt (not successfully) to reformulate it. You are asking about Kirby moves; these are associated to framed links, not links per se. So I will assume that your question is about framed links. The signature of the underlying link (in the sense of knot theory, ie as determined by a Seifert matrix) can change under Kirby moves (handle slides, in particular), so perhaps we should interpret signature to mean the signature of the linking matrix A. This is the symmetric matrix whose diagonal entries are the framings, and whose off-diagonal entries are the linking numbers between components (which are now assumed to be oriented).

However, the signature of this matrix A is not well-defined, since it changes under the blowup operation. My conclusion is that there really is no well-posed problem in a reasonable neighborhood of the one you wrote.

You ask a broader question at the end (slight rephrasing to make it a question): Is it known that there is a method to check if framed links are related by Kirby moves or not?

By Kirby's theorem, two framed links are related in this way if and only if the resulting 3-manifolds are diffeomorphic. I believe that geometrization and other recent developments imply that there is an algorithm to decide this. So in principle, there is an algorithm to decide if framed links are related by Kirby moves. However, going through this algorithm would presumably give you no insight into how to find the correct sequence of handle slides and blowups/blowdowns. Compare a similar MO thread on comparing 3-manifolds given Heegaard diagrams for both.

Finally, to answer another broad question that maybe you intended: I don't think that there are large numbers of surgery diagrams of 3-manifolds waiting for someone to check. A computational project of Lins et al produced some such diagrams, which were sorted out rather quickly by the 3-manifold community.

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