Ohtsuki invariants of the Brieskorn-Hamm manifolds and their inequalities
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.
scientific article; zbMATH DE number 6522047