We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits. Our formula for the correction term generalizes that given by Walker in his extension of Casson's SU(2) invariant to rational homology 3-spheres.

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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 (2) representations. These quantities are calculated for flat SU (2) connections on homology 3-spheres obtained by 1/k Dehn surgery on (2, q) torus knots. The methods are then applied to compute the SU (3) gauge theoretic Casson invariant (introduced in [5]) for Dehn surgeries on (2, q) torus knots for q = 3, 5, 7 and 9.

Abstract. We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians to produce minimal generating sets for singular instanton knot homology. The (symplectic) manifold where this intersection occurs corresponds to the traceless character variety of the four-punctured 2-sphere, which we identify with the familiar pillowcase. We investigate the image in this pillowcase of the traceless representations of tangles obtained by removing a trivial tangle from 2-bridge knots and torus knots. Using this, we compute the singular instanton homology of a variety of torus knots. In many cases, our computations allow us to understand nontrivial differentials in the spectral sequence from Khovanov homology to singular instanton homology.

This paper explores the relationship between the flat moduli space of a homology knot complement and other topological invariants of the knot. We define an invariant by counting the flat orbits with trace of the meridinal holonomy fixed and give a formula for the invariant in terms of the TristamLevine equivariant knot signature and the Casson invariant of the homology 3-sphere in which the knot lives.Casson defined a topological invariant, now known as the Casson invariant, for closed oriented homology 3-spheres which, roughly speaking, counts (one half) the number of irreducible representations from the fundamental group of the 3-manifold into SU(2) modulo conjugation (see [AM]). Shortly thereafter, Taubes provided an analytic definition of the same invariant, counting instead flat SU(2) connections modulo gauge equivalence (see [T]). These papers, along with related ones by Walker [Wa] and Floer [F], have inspired a long list of papers both from the topological viewpoint and from the gauge theoretic one.In 1992, Lin defined an 5 3 knot invariant by counting the number of trace-free irreducible representations of the knot group into SU(2) modulo conjugation (see [Li]). Here trace-free means that all knot meridians are taken to trace-free matrices. He then showed by a clever topological argument that this invariant equals one half of the knot signature. Ruberman suggested a generalization of this result involving equivariant knot signature which removed the trace-free condition. For the case of 2-bridge knots, a similar formula was conjectured by Heusener in [He].The aim of this article is to establish a general formula for arbitrary knots in homology 3-spheres. In the special case of knots in S 3 , this proves the formula suggested by Ruberman. Our main result states that if AS : 5 1 -> X is a smooth knot in an oriented homology 3-sphere then (under certain transversality assumptions) the number of conjugacy classes of nonabelian representations p : 7ri(X \ n) -► SU(2) taking the meridians to matrices of trace 2 cos a equals minus four times the Casson invariant of X minus one 93