Unitary representations of knot groups

Charles Frohman

doi:10.1016/0040-9383(93)90042-T

IN 1984 Casson introduced a new invariant for oriented homology three-spheres M. The invariant 1(M) is the algebraic number of irreducible W(2) representations of xi(M). The invariant reduces mod 2 to the p invariant. In the process of giving a method for computing his invariant he defined an invariant X(K) of oriented knots K in oriented homology spheres that is the algebraic number of PU(2) representations with nontrivial second Stiefel-Whitney class of the fundamental group of the result of longitudinal surgery on the knot. Casson’s invariant for knots reduces mod 2 to the arf invariant. I became interested in whether it was possible to find meaningful generalizations of Casson’s invariant by using Lie groups other than SU(2). The first step in such a program is to develop an analog of Casson’s invariant of knots. For each integer n 2 2 and each integer k there is an invariant of knots n(n, k) that gives an algebraic measure of the number of representations of the fundamental group of the result of longitudinal surgery on the knot into PU(n) so that the induced bundle on the manifold has first chern number congruent to k mod n. The definition of these invariants and the derivation of their properties is given in [4] and [S]. The material in this paper is the starting point for the computations there. In [4] it is shown that if some n(n, k) invariant is nonzero, where n and k are relatively prime then there exists an n 1 dimensional family of conjugacy classes of SU(n) representations of the fundamental group of the knot. In this paper I give a method of computing the n(n, k) invariants when K is a fibered knot and n and k are relatively prime. In this case n(n, k) is equal to a Lefschetz fixed point number defined in the body of the paper. The method of computation shows that the n(n, k) invariants of a fibered knot are determined by its Alexander polynomial. Furthermore the value of the Alexander polynomial at t = 1 and the n(n, k) invariants in turn determine the Alexander polynomial of a fibered knot in a rational homology sphere. The most striking result is Theorem 1.7. This theorem implies that if K is a fibered knot of genus g in a rational homology sphere then there exists an irreducible representation of the fundamental group of the knot complement into some SU(n) where 2 I n I g + 1. It is not unreasonable to conjecture that if K is a knot in a rational homology sphere and the Alexander polynomial of K is nontrivial, then a result similar to the one proved here for fibered knots should be true. Our method of computation is gauge theoretic in nature. It rests on the work of Atiyah and Bott [2] on Yang-Mills equations over a Riemann surface. The computation takes place in a direct summand of the cohomology of a classifying space for the gauge group of a hermitian bundle over a closed surface. Amazingly once the answer

Unitary representations of knot groups

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