Extended Alexander matrices of 3-manifolds. II: Applications
For a knot \(k\subset S^ 3\) and relatively prime integers p,q, let L(p,q,k) be the 3-manifold obtained from \(S^ 3\) by Dehn surgery along k with coefficient p/q. The author uses the ''extended Alexander matrices of 3-manifolds'' introduced in part I [see the author and \textit{J. Kanno}, ibid. 8, 107-120 (1985; Zbl 0583.57007)] to establish a necessary condition on the Alexander polynomial \(\Delta_ k(t)\) of k for L(p,q,k) to be homeomorphic to a lens space L(p,q'). Namely, if \(L(p,q,k)=L(p,q')\) and if r,r' are integers such that rq\(\equiv r'q'\equiv 1\) (mod p) then \((1+t+...+t^{r-1})\Delta_ k(t)\equiv \pm t^{\ell}u\bar u (1+t^ s+...+t^{s(r'-1)})\) mod (1\(+t+...+t^{p-1})\) for some \(u\in {\mathbb{Z}}[t]\), \(\ell,s\in {\mathbb{Z}}\) with \((p,s)=1.\) A corollary: If \(\Delta_ k=1\) and \(L(p,q,k)=L(p,q')\) then \(q\equiv \pm q'\) (mod p) or \(qq'\equiv \pm 1\) (mod p). In the case of unknotted k this yields the well-known classification of 3-dimensional lens spaces.
scientific article; zbMATH DE number 3984115