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Adjoint twisted Alexander polynomials of genus one two-bridge knots. (English) Zbl 1354.57023
The twisted Alexander polynomial is a generalization of the Alexander polynomial. It is a polynomial invariant of a \(3\)-manifold which is associated with a choice of a representation of its fundamental group.
Let \(K\subset S^3\) be a knot and \(\Pi\) the fundamental group of its exterior. Given a representation \(\rho\) of \(\Pi\) into \(SL_2(\mathbb{C})\), one can compose it with the adjoint action \(Ad\), which consists in the conjugation on the Lie algebra \(sl_2(\mathbb{C})\) by the Lie group \(SL_2(\mathbb{C})\), to obtain a representation of \(\Pi\) into \(SL_3(\mathbb{C})\).
The twisted Alexander polynomial associated to \(Ad\circ\rho\) is called the adjoint twisted Alexander polynomial associated to \(\rho\). This polynomial has been previously explicitly calculated by the author for torus knots and twist knots. In this paper, the author generalizes his computations to the family of genus one two-bridge knots, a class of two-bridge knots which includes twist knots. As a consequence, he deduces the nonabelian Reidemeister torsion of genus one two-bridge knots.

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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