Homology cylinders and sutured manifolds for homologically fibered knots
type: Item
instance of: scholarly article
title: Homology cylinders and sutured manifolds for homologically fibered knots
zbMATH Open document ID: 1287.57022
author: Hiroshi Goda
author: Takuya Sakasai
published in: Tokyo Journal of Mathematics
publication date: 2013-09-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/0807.4034
full work available at URL: https://projecteuclid.org/euclid.tjm/1374497513
reviewed by: Scott A. Taylor
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6211477
zbMATH Keywords: homology cylinder
zbMATH Keywords: homologically fibered knot
zbMATH Keywords: sutured manifold
zbMATH Keywords: Magnus representation
zbMATH Keywords: Alexander polynomial
zbMATH Keywords: torsion
zbMATH Keywords: Nakanishi index
MaRDI profile type: Publication
OpenAlex ID: W2963961938
arXiv ID: 0807.4034
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DOI: 10.3836/TJM/1374497513
review text: Both homology cylinders and sutured manifolds [\textit{D. Gabai}, J. Differ. Geom. 18, 445--503 (1983; Zbl 0533.57013)] are a particular kind of cobordism between surfaces. Informally, a homology cylinder between compact homeomorphic surfaces \(S_1\) and \(S_2\) is a 3-manifold \(M\) with \(\partial M = S_1 \cup S_2\) and with the inclusions of each of \(S_1\) and \(S_2\) into \(M\) inducing isomorphisms on integer homology groups. A sutured manifold \((M,\gamma)\), on the other hand, is a compact 3-manifold \(M\) with annuli (called sutures) \(\gamma \subset \partial M\) dividing \(\partial M\) into two surfaces \(S_1\) and \(S_2\) which are given opposite normal orientations. Each of \(S_1\) and \(S_2\) is adjacent to each component of \(\gamma\) and on each component \(A\) of \(\gamma\) the orientations of \(S_1\) and \(S_2\) induce parallel orientations on the boundary components of \(A\). Given a knot \(K \subset S^3\) and a Seifert surface \(F\) for \(K\), we can give the exterior \(S^3 - N(F)\) of \(F\) a sutured manifold structure by taking a neighborhood of \(K \subset \partial N(F)\) to be the suture. If this sutured manifold is the product of \(F\) with an interval, then \(K\) is a fibered knot. A knot is ``homologically fibered'' if the degree of its normalized Alexander polynomial \(\Delta_K(t)\) is \(2g\) where \(g\) is the Seifert genus of \(K\) and \(\Delta_K(0) = \pm 1\).NEWLINENEWLINEThe paper under review considers the relationship between homology cylinders and sutured manifolds in several different contexts.NEWLINENEWLINESections 1, 2, and 3 are introductory. Section 4 applies invariants of homology cylinders to give criteria for a homologically fibered knot to be non-fibered. The two main invariants are the ``Magnus matrix'' [\textit{T. Sakasai}, Algebr. Geom. Topol. 8, No. 2, 803--848 (2008; Zbl 1156.57001)] and the ``\(\Gamma\)-torsion' (an invariant arising from algebraic K-theory). Section 5 gives a relationship between these invariants and…
description: scientific article; zbMATH DE number 6211477