A prime decomposition theorem for \(\theta_n\)-curves in \(S^3\)
A \(\theta_n\)-curve is a graph \(\Gamma\) in the 3-sphere \(S^3\) which consists of two vertices \(v_1\), \(v_2\) and \(n\) edges, each edge joining the two vertices. A \(\theta_n\)-curve is said to be trivial if there exists a 2-sphere in \(S^3\) that contains \(\Gamma\), and otherwise nontrivial. Let \(\Sigma\) be a union of mutually disjoint 2-spheres \(\Sigma_1, \Sigma_2, \dots , \Sigma_{m-1}\) in \(S^3\). \(\Sigma\) is said to be decomposing spheres of \(\Gamma\) if \(\Sigma\) does not contain the vertices \(v_1\) and \(v_2\) of \(\Gamma\) and \(\Sigma_i\) meets each edge of \(\Gamma\) transversely at exactly one point (\(i=1, \dots , m-1\)). Let \(B_{i,j}\) (\(i=1,2\), \(j=1, \dots , m-1\)) be the 3-ball in \(S^3\) bounded by \(\Sigma_j\). We can assume that \(B_{1,1} \subset B_{1,2} \subset \dots \subset B_{1,m-1}\) by renumbering the decomposing spheres, if necessary. Let \(\Gamma_j\) be a \(\theta_n\)-curve obtained from \(\Gamma\) by contracting \(B_{1,j-1}\) and \(B_{2,j}\) to \(v_1\) and \(v_2\) respectively (\(j=1, \dots , m\)), where \(B_{1,0}\) equals \(v_1\) and \(B_{2,m}\) equals \(v_2\). \(\Gamma\) is said to be decomposed in \(\Gamma_1, \Gamma_2, \dots , \Gamma_m\) by \(\Sigma\). The main result of the article states that every nontrivial \(\theta_n\)-curve \(\Gamma\) can be decomposed into a finite number of prime \(\theta_n\)-curves \(\Gamma_1, \Gamma_2, \dots , \Gamma_m\). The \(\theta_n\)-curves \(\Gamma_i\) are uniquely determined up to order and equivalence.
scientific article; zbMATH DE number 1308035