Representing 3-manifolds in the complex number plane (Q2405090)

In previous papers the first author studied characteristic invariants for \(3\)-manifolds, which are invariants such that two \(3\)-manifolds with the same invariant are homeomorphic and can be reconstructed from the data of the invariant. In particular in [\textit{A. Kawauchi}, Kyungpook Math. J. 55, No. 4, 753--771 (2015; Zbl 1362.57008)] he obtained an embedding \(\sigma_{\alpha} :\mathbb{M}\to \mathbb{X}\), where \(\mathbb{M}\) is the set of homeomorphism types of closed connected orientable \(3\)-manifolds and \(\mathbb{X}\) is the set of integer lattice points. He defined a PDelta set \(P\bigtriangleup \subset \mathbb{X}\) and an embedding \(g:P\bigtriangleup \to [0,\infty ]\cap \mathbb{Q}\) such that \(g(P\bigtriangleup )\) recovers the set \(P\bigtriangleup\). This is used to obtain a smooth function \(G_\mathbb{S} :(-1,1)\to \mathbb{R}\), for subsets \(\mathbb{S}\subset P\bigtriangleup\). By restricting \(P\bigtriangleup\) to the subsets \(\sigma_{\alpha}(\mathbb{M})\) he obtained a characteristic invariant for \(3\)-manifolds. In the present paper an ADelta set \(A\bigtriangleup \subset \mathbb{X}\) is introduced to define an embedding \(q:A\bigtriangleup \to \mathbb{C}\), such that the norm \(|q({\mathbf x})|\leq \frac{1}{2}\) and the lattice point \({\mathbf x}\) is reconstructed from the value \(q({\mathbf x})\). By restricting \(A\bigtriangleup\) to the subsets \(\sigma_{\alpha}(\mathbb{M})\) a complex number valued characteristic invariant for \(3\)-manifolds is obtained. Then the table of the characteristic quantities of \(3\)-manifolds of lengths up to \(10\) is given and the distributive situation of the characteristic quantities in the table is plotted in the complex plane. Finally, by using this complex valued characteristic invariant, a holomorphic function on the open unit disk is constructed as a characteristic invariant for \(3\)-manifolds.