Ohtsuki invariants of the Brieskorn-Hamm manifolds and their inequalities
type: Item
instance of: scholarly article
title: Ohtsuki invariants of the Brieskorn-Hamm manifolds and their inequalities
zbMATH Open document ID: 1335.57007
author: Yasuyoshi Tsutsumi
published in: Topology and its Applications
publication date: 2015-12-18T00:00:00Z
review text: In an earlier paper [Topology Appl. 159, No. 12, 2870--2887 (2012; Zbl 1245.57006)], the author showed that the Lescop invariant of a Brieskorn-Hamm manifold is non-positive. A Brieskorn-Hamm manifold \(M=\Sigma (a_1 ,\dots,a_n)\) is a Seifert fibered \(3\)-manifold whose Seifert invariants in terms of the \(a_i\) have been computed by Neumann and Raymond. In the present paper the author considers the Brieskorn-Hamm manifolds \(M\) which are rational homology spheres. He reviews the surgery description of \(M\) and uses results obtained in his previous paper to compute \(H_1 (M)\) and the Ohtsuki invariants \(\lambda_i (M)=\lambda_i (a_1 ,\dots ,a_n )\) for \(i=0,1,2\). He then shows that \(\lambda_1 (a_1 ,\dots ,a_n )<0<\lambda_2 (a_1 ,\dots ,a_n )\), except for \(\lambda_1 (2,2,2)=0\), \(\lambda_2 (2,2,2)=-\frac{1}{64}\). Furthermore \(\lambda_2 (a_1 ,\dots ,a_n )>-\lambda_1 (a_1 ,\dots,a_n )>\lambda_0 (a_1 ,\dots ,a_n )\), except for some explicitly listed cases.
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 6522047
zbMATH Keywords: Ohtsuki invariant
zbMATH Keywords: Brieskorn-Hamm manifolds
zbMATH Keywords: Dedekind sum
reviewed by: Wolfgang Heil
MaRDI profile type: Publication
full work available at URL: j.topol.2015.05.062
OpenAlex ID: W366325695
cites work: Q3033002
cites work: Exotische Sphären als Umgebungsränder in speziellen komplexen Räumen
cites work: Q4775913
cites work: Global Surgery Formula for the Casson-Walker Invariant
cites work: Classical tessellations and three-manifolds
cites work: QuantumSO(3)-invariants dominate theSU(2)-invariant of Casson and Walker
cites work: Seiberg-Witten invariants and surface singularities
cites work: Q4185506
cites work: A polynomial invariant of rational homology \(3\)-spheres
cites work: Q4107635
cites work: Casson-Walker Invariant of Seifert Fibered Rational Homology Spheres as Quantum SO(3)-Invariant
cites work: Invariants for homology 3-spheres
cites work: Negativity of the Lescop invariants of the Brieskorn-Hamm manifolds
DOI: 10.1016/J.TOPOL.2015.05.062
description: scientific article; zbMATH DE number 6522047