KNOT AND LINK INVARIANTS AND MODULI SPACE OF PARABOLIC BUNDLES

Abstract

In this paper, we show that the representation variety of the fundamental group of a 2n-punctured S2 with different conjugacy classes in SU(2) along punctures is a symplectic stratified variety from the group cohomology point of view. The representation variety can be identified with the moduli space of s-equivalence classes of stable parabolic bundles over the 2n-punctured S2 with corresponding weights along punctures, and also can be identified with the moduli space of gauge equivalence classes of SU(2)-flat connections with prescribed holonomies along punctures. We obtain an invariant of links (knots) from intersection theory on such a moduli space (a generalization of the signature of the link). We also study a SL2(C)-character variety of a knot K in S3 with fixed holonomy μ + μ-1 along the meridian of π1(S3\ K) (μ ∈ C*). The fixed-trace condition rules out the possibility of reducible representations with non-abelian image and the ideal point of irreducible representations via a generic perturbation. For knots without closed incompressible surfaces in S3 \ K, we show that there is a well-defined SL2(C)-knot invariant which is also related to them A-polynomial for special values μ ∈ U(1)\ {± 1}.