On the slice-ribbon conjecture for Montesinos knots

Abstract

We establish the slice-ribbon conjecture for a family P \mathscr {P} of Montesinos knots by means of Donaldson’s theorem on the intersection forms of definite 4 4 -manifolds. The 4 4 -manifolds that we consider are obtained by plumbing disc bundles over S 2 S^2 according to a star-shaped negative-weighted graph with 3 3 legs such that: i) the central vertex has weight less than or equal to − 3 - 3 ; ii) − total weight − 3 # vertices > − 1 -\,\mbox {total weight} - 3\, \# \mbox {vertices} >-1 . The Seifert spaces which bound these 4 4 -dimensional plumbing manifolds are the double covers of S 3 S^3 branched along the Montesinos knots in the family P \mathscr {P} .