Extended Alexander matrices of 3-manifolds. II: Applications
type: Item
instance of: scholarly article
title: Extended Alexander matrices of 3-manifolds. II: Applications
zbMATH Open document ID: 0608.57004
author: Shinji Fukuhara
published in: Tokyo Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 3984115
zbMATH Keywords: knot
zbMATH Keywords: Dehn surgery
zbMATH Keywords: Alexander matrices
zbMATH Keywords: Alexander polynomial
zbMATH Keywords: lens space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270151230
OpenAlex ID: W2041392398
DOI: 10.3836/TJM/1270151230
Recommended article: Extended Alexander matrices of 3-manifolds. I
Recommended article: The \(L^2\)-Alexander torsions of 3-manifolds
Recommended article: The \(L^2\)-Alexander torsion of 3-manifolds
Recommended article: The Alexander polynomial and finite type 3-manifold invariants
Recommended article: Twisted Alexander polynomials and fibered 3-manifolds
Recommended article: Manifolds of Alexander type
Recommended article: The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology1
Recommended article: A combinatorial matrix in 3-manifold theory
Recommended article: On the Alexander polynomials of open book decompositions of 3-manifolds
Recommended article: Alexandroff type manifolds and homology manifolds
description: scientific article; zbMATH DE number 3984115