Three-Dimensional Representations of Punctured Torus Bundles

Abstract

In this paper, we construct a complex curve of irreducible [Formula: see text] representations of the fundamental group of a once punctured torus bundle over the circle. These representations are different from those obtained by composing representations in [Formula: see text] with the unique irreducible representation of [Formula: see text] in [Formula: see text]. Moreover, infinitely many of these representations are conjugate to SU(3) representations. We conclude the paper with a computation of the curve in the case that the bundle is the figure-eight knot complement, and we show that for infinitely many Dehn surgeries on the figure-eight knot, there is a representation from this curve that descends to a representation of the fundamental group of the surgered manifold.