\(\text{SO}_ 3(\mathbb{R})\)-representation curves for two-bridge knot groups
Given a 2-bridge knot or link group, the corresponding SO\(_ 3(\mathbb{R})\) representation space may be described by a real plane algebraic curve. These real algebraic curves are closely related to the nab-rep curves investigated by \textit{R. Riley} [Q. J. Math., Oxf. II. Ser. 35, 191-208 (1984; Zbl 0549.57005)]. The study of the geometry of the representation curves raise in its first step the question of reducibility. It was a conjecture of R. Riley that the representation curve of a 2-bridge knot group has extra algebraic components if the knot admits an unusually large group of symmetries which preserve both space and knot orientation. The next question which one might ask is what is the genus of the representation curve. It was a conjecture of \textit{G. Burde} [Math. Ann. 288, 103-119 (1990; Zbl 0694.57003)] that the representation curves split always into rational components. The aim of this paper is to give an affirmative answer to Riley's conjecture and to present an example of an irreducible non-rational representation curve which disproves Burde's conjecture. Moreover, the last section connects (in special cases) Casson's invariant and the signature of knots with the complexification of the representation curves.
SO\(_ 3(\mathbb{R})\) representation space
complexification of the representation curves
scientific article; zbMATH DE number 537320
scientific article