On the generalization of union of knots (Q789064)

A generalization of the sum of two knots \(k_ 1\) and \(k_ 2\) in \(S^ 3\) is defined as follows: \(k_ 1+_ n k_ 2\) consists of the class of knots having a representative which is the usual sum of \(k_ 1\) and \(k_ 2\) modified by ''clasping'' a pair of subarcs of the sum along an arc which meets the separating 2-sphere of the sum in n points; for \(n=1\) this is the Kinoshita-Terasaka union of knots. The point of the definition is to attack the well-known conjecture that knots with unknotting number 1 are prime; indeed, this conjecture would follow from the truth of the conjecture that the trivial knot cannot be obtained as a member of \(k_ 1+_ n k_ 2\) for nontrivial knots \(k_ 1\) and \(k_ 2\). The latter conjecture is known to be true for \(n=0\) and \(n=1\); this paper provides a proof for the case \(n=2\) following a cut-and-paste argument similar to \textit{H. Terasaka}'s proof of the case \(n=1\) [Osaka Math. J. 12, 113-144 (1960; Zbl 0118.392)].