Uniqueness of minimal genus Seifert surfaces for links (Q1825474)

The author studies the problem suggested by the title. Let \(L_ 1\), \(L_ 2\) be links with minimal genus Seifert surfaces \(R_ 1\), \(R_ 2\). Let R be a Murasugi sum of \(R_ 1\) and \(R_ 2\). The author proves the following: The minimal genus Seifert surfaces for \(L=\partial R\) are unique if and only if one of the \(L_ i\) is fibered, and the minimal genus Seifert surfaces for the other are unique. Knots with non-unique, minimal genus Seifert surfaces with fewer than 11 crossing are listed.