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On reciprocality of twisted Alexander invariants. (English) Zbl 1200.57005
By a classical result, the Alexander polynomial of a knot is reciprocal, due to duality of presentations or spaces, depending of one’s point of view. In the presence of an $$n$$-dimensional linear representation $$\gamma : \pi \rightarrow GL_nR$$ of the knot group $$\pi$$ we have twisted polynomial invariants, and the paper goes into the question of reciprocality of those. By Theorem 3.2, the twisted Reidemeister torsion $$\tau_\gamma(t)$$ and the twisted Alexander polynomial $$\Delta_\gamma(t)$$ are reciprocal, provided $$R={\mathbb C}$$ and $$\gamma$$ is conjugate to its dual (inverse-transpose). For $$R={\mathbb Z}$$, a representation into $$SL_3{\mathbb Z}$$ is presented such that $$\tau_\gamma(t)$$ is non-reciprocal.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
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