The dual knots of doubly primitive knots

Abstract

For certain $(1,1)$-knots in lens spaces with a longitudinal surgery yielding the 3-sphere, we determine a non-negative integer derived from its $(1,1)$-splitting. The value will be an invariant for such knots. Roughly, it corresponds to a `minimal' self-intersection number when one consider projections of a knot on a Heegaard torus. As an application, we give a necessary and sufficient condition for such knots to be hyperbolic.