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Hillman, Jonathan A.; Silver, Daniel S.; Williams, Susan G.

On reciprocality of twisted Alexander invariants. (English) Zbl 1200.57005

Algebr. Geom. Topol. 10, No. 2, 1017-1026 (2010).
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.
Reviewer: Dieter Erle (Dortmund)

Cited in 8 Documents

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.

Keywords:

reciprocal polynomial; twisted Reidemeister torsion; twisted Alexander polynomial; linear representation; knot group
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\textit{J. A. Hillman} et al., Algebr. Geom. Topol. 10, No. 2, 1017--1026 (2010; Zbl 1200.57005)
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