A relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion

Yamaguchi, Yoshikazu

Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 337-362.

Nous montrons une relation entre la torsion de Reidemeister non-acyclique et un zéro de la torsion de Reidemeister acyclique pour une représentation $\lambda$-régulière dans $\mathrm{SU}\left(2\right)$ ou $\mathrm{SL}(2,ℂ)$ du groupe d’un nœud. Alors nous pouvons donner une méthode pour calculer la torsion de Reidemeister non-acyclique de l’extérieur d’un nœud. Nous calculons un nouvel exemple et étudions le comportement de la torsion de Reidemeister non-acyclique associée à un nœud à deux-ponts et une $\mathrm{SU}\left(2\right)$-représentations du groupe du nœud.

We show a relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion for a $\lambda$-regular $\mathrm{SU}\left(2\right)$ or $\mathrm{SL}(2,ℂ)$-representation of a knot group. Then we give a method to calculate the non-acyclic Reidemeister torsion of a knot exterior. We calculate a new example and investigate the behavior of the non-acyclic Reidemeister torsion associated to a $2$-bridge knot and $\mathrm{SU}\left(2\right)$-representations of its knot group.

Classification : 57Q10, 57M05, 57M27
Keywords: Reidemeister torsion, twisted Alexander invariant, knots, representation spaces
Mot clés : torsion de Reidemeister, invariant tordu de Alexander, nœuds, variétés des représentations

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