Cassonʹs knot invariant and gauge theory

It is known that twice the Casson invariant for integral homology 3 spheres is equal to the Euler characteristic of the Floer homology group of them. Here we show that a similar result holds in case of the Casson invariant for knots in integral homology 3 spheres. This result is obtained as a corollary of Floerʹs exact triangle. But we give a more elementary proof here. We also show that a similar result holds in case of the Casson–Walker invariant for null homologous knots in rational homology 3 spheres. This result is not obtained as a corollary of Floerʹs exact triangle, and so is new. These results will serve as a starting point to obtain the Dehn surgery formula for the Floer homology groups of general 3-dimensional manifolds.