# Two links with the same signatures but unknown if they are related by Kirby moves

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.

• 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. – Danny Ruberman Aug 28 '13 at 2:41
• @DannyRuberman Yes, these links should be framed links. Thanks. – user22741 Aug 28 '13 at 2:56