Variation of the Alexander?Conway polynomial under Dehn surgery

Abstract

Let ε 1 and ε 2 belong to {±1}. When the ε 1 -surgery along a knot K 1 in S 3 produces the same homology sphere as the ε 2 -surgery along a knot K 2 in S 3 , then the Casson surgery formula implies that ε 1 Δ K 1 ″(1)= ε 2 Δ K 2 ″(1), where Δ ( t ) denotes the symmetric Alexander polynomial. For any pair ( Λ 1 ( t ), Λ 2 ( t )) of possible knot Alexander polynomials such that ε 1 Λ 1 ″(1)= ε 2 Λ 2 ″(1), we exhibit a pair ( K 1 , K 2 ) of knots in S 3 such that Δ K 1 ( t )= Λ 1 ( t ), Δ K 2 ( t )= Λ 2 ( t ) and the ε 1 -surgery along K 1 produces the same homology sphere as the ε 2 -surgery along K 2 .