On the kernel of the Casson invariant
This paper is devoted to proving that if two integral homology spheres have equal Casson invariants, then one can be obtained from the other by surgery on a \(\pm 1\)-framed boundary link, each of whose components has Alexander polynomial \(1\). This is seen to be the same as saying that one can be obtained from the other by a sequence of surgeries, each on a \(\pm 1\)-framed knot with Alexander polynomial \(1\). The argument begins with the observation that we can always go from one integral homology sphere to any other by a sequence of surgeries on \(\pm 1\)-framed knots. By introducing band crossings in the Seifert surfaces of these knots one can arrange that they all have Alexander polynomial \(1\). Then one has to deal with the circles enclosing the bands on which \(\pm 1\)-surgery creates the band-crossings. These are knots of genus one. The main part of the argument is to show that, when the Casson invariants of the two homology spheres are equal, these knots can be paired up so that surgery on any pair can be replaced by a sequence of surgeries on Alexander polynomial \(1\) knots.
scientific article; zbMATH DE number 1089981