A relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion
[Une relation entre la torsion de Reidemeister non acyclique et un zéro de la torsion de Reidemeister acyclique]
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 λ-régulière dans SU (2) ou SL (2,) 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 SU (2)-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 λ-regular SU (2) or SL (2,)-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 SU (2)-representations of its knot group.

DOI : 10.5802/aif.2352
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
Yamaguchi, Yoshikazu 1

1 University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba Meguro Tokyo 153-8914 (Japan)
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Yamaguchi, Yoshikazu. A relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion. Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 337-362. doi : 10.5802/aif.2352. http://archive.numdam.org/articles/10.5802/aif.2352/

[1] Brown, K. S. Cohomology of Groups, Graduate Texts in Mathematics 87, Springer-Verlag, New York, 1994 | MR | Zbl

[2] Burde, G. SU (2)-representation spaces for two-bridge knot groups, Math. Ann., Volume 288 (1990), pp. 103-119 | DOI | MR | Zbl

[3] Burde, G.; Zieschang, H. Knots ( Second edition ) , de Gruyter Studies in Mathematics 5, Walter de Gruyter, 2003 | MR | Zbl

[4] Dubois, J. Non abelian Reidemeister torsion and volume form on the SU (2)-representation space of knot groups, Ann. Inst. Fourier, Volume 55 (2005), pp. 1685-1734 | DOI | Numdam | MR | Zbl

[5] Dubois, J. Non abelian twisted Reidemeister torsion for fibered knots, Canad. Math. Bull., Volume 49 (2006), pp. 55-71 | DOI | MR | Zbl

[6] Dubois, J.; Kashaev, R. On the asymptotic expansion of the colored Jones polynomial for torus knots (to appear in Math. Ann., arXiv:math.GT/0510607)

[7] Heusener, M.; Klassen, E. Deformations of dihedral representations, Proc. Amer. Math. Soc., Volume 125 (1997), pp. 3039-3047 | DOI | MR | Zbl

[8] Kirk, P.; Klassen, E. Chern-Simons invariants of 3-manifolds and representation spaces of knot groups, Math. Ann., Volume 287 (1990), pp. 343-367 | DOI | MR | Zbl

[9] Kirk, P.; Livingston, C. Twisted Alexander Invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology, Volume 38 (1999), pp. 635-661 | DOI | MR | Zbl

[10] Kitano, T. Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math., Volume 174 (1996), pp. 431-442 | MR | Zbl

[11] Klassen, E. Representations of knot groups in SU (2), Trans. Amer. Math. Soc., Volume 326 (1991), pp. 795-828 | DOI | MR | Zbl

[12] Milnor, J. Whitehead torsion, Bull. Amer. Math. Soc., Volume 72 (1966), pp. 358-426 | DOI | MR | Zbl

[13] Milnor, J. Infinite cyclic coverings, Conference on the Topology of Manifolds, Prindle Weber & Schmidt Boston, Mass., Michigan State Univ. 1967 (1968), pp. 115-133 | MR | Zbl

[14] Morgan, J. W.; Shalen, P. B. Valuations, trees, and degenerations of hyperbolic structures, Ann. of Math. (2), Volume 120 (1984), pp. 401-476 | DOI | MR | Zbl

[15] Porti, Joan Torsion de Reidemeister pour les variétés hyperboliques, Mem. Amer. Math. Soc., Volume 128 (1997) no. 612, pp. x+139 | MR | Zbl

[16] Riley, R. Nonabelian representations of 2-bridge knot groups, Quart. J. Math. Oxford Ser. (2), Volume 35 (1984), pp. 191-208 | DOI | MR | Zbl

[17] Rolfsen, D. Knots and links, Mathematics Lecture Series 7, Publish or Perish Inc., Houston, TX, 1990 | MR | Zbl

[18] Spanier, E. H. Algebraic Topology, Springer-Verlag, New York-Berlin, 1981 | MR | Zbl

[19] Turaev, V. Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001 | MR | Zbl

[20] Turaev, V. Torsions of 3 -dimensional manifolds, Progress in Mathematics 208, Birkhäuser Verlag, Basel, 2002 | MR | Zbl

[21] Wada, M. Twisted Alexander polynomial for finitely presentable groups, Topology, Volume 33 (1994), pp. 241-256 | DOI | MR | Zbl

[22] Yamaguchi, Y. Limit values of the non-acyclic Reidemeister torsion for knots (arXiv:math.GT/0512277)

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