On the 1-bridge genus of small knots

A Heegaard splitting for S3 gives a 1-bridge presentation for a knot k⊂S3 if the knot intersects each handlebody of the Heegaard splitting in an arc which forms the interior part of the boundary of a disk in the handlebody. The minimal g for which the knot has a 1-bridge presentation of genus g is called the 1-bridge genus g1(k). The object of the article is the behaviour of this invariant under the connected sum k1#k2. More precisely, for small knots k1 and k2 which are knots which do not have essential closed surfaces in their exteriors, our purpose is to show the following inequality: g1(k1)+g1(k2)−1≤g1(k1#k2)≤g1(k1)+g1(k2).