Casson's knot invariant and gauge theory

✍ Scribed by K. Masataka

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 250 KB
• Volume: 112
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(99)00230-8

✦ Synopsis

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 Floer's 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 Floer's 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.