An orientation for the SU(2)-representation space of knot groups
Let \(K\subset S^3\) be a knot and denote by \(\hat R(K)\) the space of equivalence classes of irreducible representations of the knot group \(G=\pi_1(S^3\setminus K)\) in \(SU(2)\). Let \(Reg(K)\subset \hat R(K)\) be the space of regular representations, i.e irreducible representations \(\rho:G\rightarrow SU(2)\) such that \(H_\rho^1(G)\cong rb\) where \(H_\rho^*(G)=H^*(G,Ad\rho)\) denotes the twisted cohomology group of \(G\) with coefficients in \(su(2)\). It was proved by \textit{M. Heusener} and \textit{E. P. Klassen} in [Proc. Am. Math. Soc. 125, 3039-3047 (1997; Zbl 0883.57001)] that \(Reg(K)\subset \hat R(K)\) is a real one dimensional manifold. In this paper, the author proves that \(Reg(K)\) also carries a canonical orientation. A first consequence of this result is that there exists a large class of knots which have at least a one dimensional representation space \(\hat R(K)\). As a further application, the author explains a generalization of a result of \textit{X. S. Lin} [J. Differential Geometry 35, 337-357 (1992; Zbl 0774.57007)] \textit{M. Heusener} and \textit{J. Kroll} [Comment. Math. Helv. 73, 480-498 (1998; Zbl 0910.57004)]. Using the orientation of the representation space, he proves that the generalized Lin invariant and the signature of the knot are the same quantities as they turn out to be a weighted sum of zeros of the Alexander polynomial on the unit circle.
scientific article; zbMATH DE number 1901934
scientific article