Simple ribbon fusions and genera of links

Abstract

Let $K$ be the result of a 1-fusion (band sum) of a knot $k$ and a distant trivial knot in $S^3$. From results of D. Gabai and of M. G. Scharlemann, we know that the genus of $K$ is at least that of $k$ and that equality holds if and only if the band sum is, in fact, a connected sum (in which case $K$ is ambient isotopic to $k$). In this paper, we consider a generalization of this result to an $m$-fusion of a link and a distant trivial link with $m$-components.