This paper provides infinitely many supporting examples for a conjecture by D. Gang, S. Kim, and the author [D. Gang et al., Adv. Theor. Math. Phys. 25, No. 7, 1819–1845 (2021; Zbl 1498.81105)]. They conjectured that if the \(\mathrm{SL}_2(\mathbb{C})\)-character variety for a hyperbolic \(3\)-manifold with boundary consists of \(1\)-dimensional components then the inverse sum of the adjoint Reidemeister torsion for any slope on the boundary torus equals zero. Here the inverse sum of the adjoint Reidemeister torsion runs over a generic level set of the trace function of a given slope on the \(\mathrm{SL}_2(\mathbb{C})\)-character variety. The supporting examples of this paper are given by hyperbolic two-bridge knot exteriors and their meridian slopes. This paper calls the vanishing sum the vanishing identity of the adjoint Reidemeister torsion.
This paper has two main results. In the first main result, the adjoint Reidemeister torsion for a hyperbolic two-bridge knot exterior and the meridian slope is expressed as a rational function on the \(\mathrm{SL}_2(\mathbb{C})\)-character variety. That function is given by the defining polynomial of the character variety, called the Riley polynomial of a two-bridge knot. The author also shows that the inverse sum of the adjoint Reidemeister torsion on a generic level set of the trace function of the meridian turns into zero as the second main result. The second result is derived from the Euler-Jacobi theorem when we think of the adjoint Reidemeister torsion as a rational function on the \(\mathrm{SL}_2(\mathbb{C})\)-character variety for a hyperbolic two-bridge knot exterior. This paper also touches on the inverse sum of the adjoint Reidemeister torsion for non-hyperbolic two-bridge knots and the meridian slope. According to a numerical computation by the author, the inverse sum for a non-hyperbolic two-bridge knot over a generic level set should be \(\pm 2\).