The Kakimizu complex of a connected sum of links (Q2849039)

A Seifert surface of a link \(L\) is said to be taut if it has maximal Euler characteristic among all Seifert surfaces of \(L\). The Kakimizu complex, \(\mathrm{MS}(L)\), of \(L\) was originally defined in [\textit{O. Kakimizu}, Hiroshima Math. J. 22, No. 2, 225--236 (1992; Zbl 0774.57006)] as the the simplicial complex whose vertices are the ambient isotopy classes of the taut Seifert surfaces of \(L\), and in which \(n\) distinct vertices span an \(n\)-simplex exactly when they can be realised disjointly. Kakimizu proved that if \(K_1\) and \(K_2\) are each non-fibred knots with a unique incompressible Seifert surface then \(|\mathrm{MS}(L)|\) is homeomorphic to \(\mathbb{R}\).NEWLINENEWLINEIn [\textit{P. Przytycki} and \textit{J. Schultens}, Trans. Am. Math. Soc. 364, No. 3, 1489--1508 (2012; Zbl 1239.57025)] the Kakimizu complex was redefined in a way that allows for the distance between two vertices to be calculated in the infinite cyclic cover of a covering space. This redefined complex, \(\mathrm{MS}'(L)\), agrees with \(\mathrm{MS}(L)\) when \(L\) does not have disconnected Seifert surfaces. The main result of the paper under review is that when \(L_1\) and \(L_2\) are non-split, non-fibred links, then \(|\mathrm{MS}'(L_1\# L_2)|\) is homeomorphic to \(|\mathrm{MS}'(L_1)|\times |\mathrm{MS}'(L_2)|\times\mathbb{R}\). In addition, an analogous result is obtained for the the case where incompressible Seifert surfaces are used instead of taut ones.