Flat connections, the Alexander invariant, and Casson's invariant
In 1992, motivated by the definition of the Casson invariant, X. S. Lin defined an invariant of knots in \(S^3\) by counting, with appropriately defined signs, the number of conjugacy classes of \(SU(2)\)-representations of the group of the knot, whose values on any meridian element have zero trace. Then Lin proved that this invariant is, in fact, determined by the signature of the knot. The aim of the present work is to generalize Lin's results in two directions. First of all it considers a count of representations whose values on any meridian have trace \(2\cos\alpha\), for any specified \(\alpha\). Secondly it considers knots in any homology \(3\)-sphere. These generalized definitions are made using the gauge theory treatment of the Casson invariant initiated by Taubes and previous work of the author on \(SU(2)\)-representations of knot groups. It is then shown that this invariant, for a knot \(K\subset X\) and specified \(\alpha\), is given by the formula \(-4\lambda(X)-\textstyle{1\over 2}\sigma_{2\alpha}(K) \), for any \(0<\alpha <\pi\) for which \(e^{2i\alpha}\) is not a root of the Alexander polynomial of \(K\). \(\lambda\) denotes the Casson invariant and \(\sigma_{2\alpha}\) denotes the twisted signature which is defined as the signature of the skew-Hermitian matrix \(e^{i\alpha}V-e^{-i\alpha}V^t\) for any Seifert matrix \(V\) of \(K\). The case \(\alpha =\pi /2\) gives Lin's results.
scientific article; zbMATH DE number 1060138