Variation of the Alexander-Conway polynomial under Dehn surgery |
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| Authors: | Yukihiro Tsutsumi Harumi Yamada |
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| Institution: | a Department of Mathematics, Faculty of Science and Technology, Keio University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522, Japan
b Department of Mathematics, School of Education, Waseda University, Nishi-Waseda 1-6-1, Shinjuku-ku, Tokyo 169-8050, Japan |
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| Abstract: | Let ε1 and ε2 belong to {±1}. When the ε1-surgery along a knot K1 in S3 produces the same homology sphere as the ε2-surgery along a knot K2 in S3, then the Casson surgery formula implies that ε1ΔK1″(1)=ε2ΔK2″(1), where Δ(t) denotes the symmetric Alexander polynomial. For any pair (Λ1(t),Λ2(t)) of possible knot Alexander polynomials such that ε1Λ1″(1)=ε2Λ2″(1), we exhibit a pair (K1,K2) of knots in S3 such that ΔK1(t)=Λ1(t), ΔK2(t)=Λ2(t) and the ε1-surgery along K1 produces the same homology sphere as the ε2-surgery along K2. |
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| Keywords: | Alexander polynomial Conway polynomial Dehn surgery Casson invariant |
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