Artin presentations of complex surfaces (Q2493397)

An Artin presentation on \(n\) generators is a group presentation \(r\) with generators \(x_1,\dots, x_n\) and relators \(r_1,\dots, r_n\) such that \(x_1\cdots x_n=(r_1^{-1} x_1 r_1)\cdots (r_n^{-1} x_n r_n)\). The set of all such presentations forms a group \(R_n\) isomorphic to \(P_n\times \mathbb Z^n\), the group of framed pure braid on \(n\) strings. The fundamental group of a closed orientable \(3\)-manifold admits an Artin presentation. Conversely, any element \(r\in R_n\) determines such a \(3\)-manifold \(M(r)\) bounding a smooth compact \(1\)-connected \(4\)-manifold \(W(r)\). The authors focus on the situation \(M(r)=S^3\). In that case, by adding a \(4\)-handle one obtains a closed \(1\)-connected \(4\)-manifold \(X(r)=W(r)\cup D^4\). The complex elliptic surfaces \(E(n)\) are an important class of such \(4\)-manifolds. The purpose of the article is to construct their associated Artin presentations. Starting from known surgery diagrams on framed links for \(E(n)\) the authors derive the corresponding framed pure braids, which in turn determine the Artin presentations. An algorithm that achieves the last step is given for an arbitrary element in \(P_n\times \mathbb Z^n\), but explicit computations are carried out only for the \(K3\) surface.