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A deformation of the Alexander polynomials of knots yielding lens spaces

Published online by Cambridge University Press:  17 April 2009

Teruhisa Kadokami  and
Yuichi Yamada
Teruhisa Kadokami
Affiliation:
Department of Mathematics, Osaka City University, Sugimoto3–3–138, Sumiyoshi-ku, Osaka 558–8585, Japan, e-mail: kadokami@sci.osaka-cu.ac.jp
Yuichi Yamada
Affiliation:
Department of Systems Engineering, The University of Electro-Communications, 1–5–1 Chofu-gaoka, Chofu, tokyo 182–8585, Japan, e-mail: yyyamada@sugaku.e-one.uec.ac.jp
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For a knot K in a homology 3-sphere Σ, by Σ(K;p/q), we denote the resulting 3-manifold of p/q-surgery along K. We say that the manifold or the surgery is of lens type if Σ(K;p/q) has the same Reidemeister torsion as a lens space.

We prove that, for Σ(K;p/q) to be of lens type, it is a necessary and sufficient condition that the Alexander polynomial ΔK(t) of K is equal to that of an (i, j)-torus knot T(i, j) modulo (tp – 1).

We also deduce two results: If Σ(K;p/q) has the same Reidemeister torsion as L(p, q') then (1) (2) The multiple of ΣK(tk) over k ∈ (i) is ±tm modulo (tp – 1), where (i) is the subgroup in (Z/pZ)×/{±1} generated by i. Conversely, if a subgroup H of (Z/pZ)×/{±l} satisfying that the product of ΣK(tk) (k ∈ H) is ±tm modulo (tp – 1), then H includes i or j.

Here, i, j are the parameters of the torus knot above.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 75 , Issue 1 , February 2007 , pp. 75 - 89
Copyright
Copyright © Australian Mathematical Society 2007

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