Gauge theoretic invariants of Dehn surgeries on knots

Abstract

New methods for computing a variety of gauge theoretic invariants for homology 3–spheres are developed. These invariants include the Chern–Simons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible $SU\left(2\right)$ representations. These quantities are calculated for flat $SU\left(2\right)$ connections on homology 3–spheres obtained by $1∕k$ Dehn surgery on $\left(2,q\right)$ torus knots. The methods are then applied to compute the $SU\left(3\right)$ gauge theoretic Casson invariant (introduced in [J. Diff. Geom. 50 (1998) 147-206]) for Dehn surgeries on $\left(2,q\right)$ torus knots for $q=3,5,7$ and 9.