Abstract
A Heegaard splitting for S 3 gives a 1-bridge presentation for a knot k⊂ S 3 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 g 1( k). The object of the article is the behaviour of this invariant under the connected sum k 1# k 2. More precisely, for small knots k 1 and k 2 which are knots which do not have essential closed surfaces in their exteriors, our purpose is to show the following inequality: g 1( k 1)+ g 1( k 2)−1≤ g 1( k 1# k 2)≤ g 1( k 1)+ g 1( k 2).
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