An intersection homology invariant for knots in a rational homology 3-sphere

Abstract

THE purpose of this paper is to define a family of computable homological invariants of knots that generalize Casson’s invariant of knots. Let K be a homologically trivia1 knot in a rational homology 3-sphere N. Given a pair of integers (n, d) with n 2 1, we define a numerical invariant & and a related polynomial invariant pJt) which depend only on (N, K, n, d mod n). The invariant An,* can be thought of as an algebraic count of the number of characters of representations of the fundamental group of the complement of K into the Lie group SU(n) which take a longitude to eznid’” times the identity. The case where n and d are not relatively prime is of most interest to us here as the relatively prime case (for fibered knots) has been treated in [4]. While we define the invariants d see Theorems 5.21 and 5.22. Furthermore, an algorithm is given for determining these polynomials. We explicitly evaluate 1,. 0; see Theorem 6.4. For fibered knots, A,,, can be computed from the intersection homology Lefschetz number of the monodromy action on the moduli space of semistable holomorphic bundles of rank n and degree d and fixed determinant over a compact Riemann surface. For n and d not relatively prime, this moduli space is typically singular. Our computation relies heavily on the theory developed by Frances Kirwan ([ 14, 15, 16, 171) for desingularizing these spaces and computing their (mid-perversity) intersection homology. The polynomial invariants which we define in $3 are best understood in the abstract framework of “cobordism functors” developed in $1. The axioms for such functors are somewhat reminiscent of, albeit less restrictive than, the axioms proposed for so-called “topological quantum field theories” (see Cl]). In [6] it was shown how the Alexander polynomial arises in an elementary fashion from U(1) representations in the context of cobordism functors (see [6, Theorem 4.41). Here, this is generalized to PU(n) representations from which we can obtain polynomial invariants which, at least in the case of fibered knots, are computable in terms of data derived from the Alexander polynomial. In order to put our theory into perspective, we first review the definition of Casson’s invariant. Let M be an oriented homology 3-sphere. Let H1 and Hz be two handlebodies so