Adjoint twisted Alexander polynomials for parabolic representations of genus one two-bridge knots

抄録

In previous papers [19, 17], we computed the twisted Alexander polynomials and adjoint twisted Alexander polynomials for nonabelian SL₂(C)-representations of genus one two-bridge knots. In this paper, we investigate their properties. When the SL₂2(C)-representation of a genus one two-bridge knot is parabolic, we show that

(1) the coefficient of the highest degree term of the adjoint twisted Alexander polynomial is a rational multiple, independent of the representation, of the square of the twisted Alexander polynomial at t = 1, and

(2) the adjoint twisted Alexander polynomial is a monic polynomial if and only if the knot is fibered.