Splitting the spectral flow and the Alexander matrix

Abstract

This paper is concerned with a procedure for computing the spectral flow of a path of self-adjoint operators of the form D, = *dA, -dA,*, where the At are SU(2) connections on a 3-manifold Z which is split along a torus, and A0 and A~ are fiat. Recent theorems of Yoshida [Y1, Y2] show how to carry this out when Z is obtained by surgery on a knot, under certain nondegeneracy conditions. Under the assumption that there is a path A, of flat connections on the knot complement and that the space of flat connections modulo gauge transformation is a smooth l-dimensional variety near this path, Yoshida shows with an explicit formula that the spectral flow is determined by the restriction of the path to the boundary torus. As a consequence of our main result we show that when the path A, has singularities, the spectral flow is not determined by its restriction to the boundary torus. We give explicit computations in w comparing the spectral flow on a surgery of a Whitehead double of a knot to the spectral flow on the corresponding surgery of the Whitehead double of the unknot. These examples have paths of flat connections on the knot complements whose restrictions to the boundary are the same, while their spectral flows differ. Suppose Z = X w Y, where X is the complement of a knot in S 3. Let A0 and Al be flat connections on Z whose restrictions to X are reducible. Then there are corresponding flat connections A ~ and At on Z ' = X ' u Y, where X' is the unknot complement (i.e., a solid torus). In ~4 we show that the difference between the spectral flow from A~ to AI on Z ' and the spectral flow from Ao to A~ on Z is a classical knot signature, and in fact is equal to the spectral flow of the Alexander matrix of the knot. Applying this theorem to satellite knots yields examples in