A Casson-lin Type Invariant for Links Eric Harper and Nikolai Saveliev

One of the characteristic features of the fundamental group of a closed 3manifold is that its representation variety in a compact Lie group tends to be finite, in a properly understood sense. This has been a guiding principle for defining invariants of 3-manifolds ever since Casson defined his λ-invariant for integral homology 3-spheres via a signed count of the SU(2) representations of the fundamental group. The signs were determined using Heegaard splitting. Among numerous generalizations of Casson’s construction, we will single out the invariant of knots in S3 defined by X.-S. Lin [8] via a signed count of SU(2) representations of the fundamental group of the knot exterior. The latter is a 3-manifold with non-empty boundary so the above finiteness principle only applies after one imposes a proper boundary condition. Lin’s choice of the boundary condition, namely, that all the knot meridians are represented by trace-free SU(2) matrices, resulted in an invariant h(K) of knots K ⊂ S3. Lin further showed that h(K) in fact equals half the knot signature of K. The signs in Lin’s construction were determined using braid representations for knots. Heusener and Kroll [7] extended this construction by letting the meridians of the knot be represented by SU(2) matrices with a fixed trace which need not be zero. Their construction gives, for each choice of the trace, a knot invariant which turns out to equal one half times the equivariant knot signature.