A study on the moduli space of representations and the twisted Alexander invariant

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The purpose of this research was to clarify fundamental properties of the twisted Alexander invariant (TAI) as a function on the moduli space of SL(2,C)-representations of the fundamental group and to give a framework for deriving geometric information of 3-manifolds from the TAI. The results are as follows.(1) We gave explicit formulas of degrees of the TAI and the defining equation of the moduli space for an infinite sequence of knots which is a generalization of twist knots. As an application, we described a necessary and sufficient condition that these knots are fibered as a kind of finiteness theorems.(2) For torus knots, we gave an explicit formula and a characterization of the TAI as a function on the moduli space of SL(2,C)-representations.