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We give a rational surgery formula for the Casson-Walker invariant of a2-component link in $S^{3}$ which is a generalization of Matveev-Polyak'sformula. As application, we give more examples of non-hyperbolic L-space $M$such that knots in $M$ are determined by their complements. We also apply theresult for the cosmetic crossing conjecture.

Applications of the Casson-Walker invariant to the knot complement and the cosmetic crossing conjectures