Algebraic & Geometric Topology

The twisted Alexander polynomial for finite abelian covers over three manifolds with boundary

Jérôme Dubois and Yoshikazu Yamaguchi

Full-text: Open access


We provide the twisted Alexander polynomials of finite abelian covers over three-dimensional manifolds whose boundary is a finite union of tori. This is a generalization of a well-known formula for the usual Alexander polynomial of knots in finite cyclic branched covers over the three-dimensional sphere.

Article information

Algebr. Geom. Topol., Volume 12, Number 2 (2012), 791-804.

Received: 3 September 2011
Revised: 20 December 2011
Accepted: 3 January 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds

Reidemeister torsion Twisted Alexander polynomial Branched cover Links Homology orientation


Dubois, Jérôme; Yamaguchi, Yoshikazu. The twisted Alexander polynomial for finite abelian covers over three manifolds with boundary. Algebr. Geom. Topol. 12 (2012), no. 2, 791--804. doi:10.2140/agt.2012.12.791.

Export citation


  • M Hirasawa, K Murasugi, On the twisted Alexander Polynomials of Knots, from: “Proceedings of Hakone Seminar on Graphs and 3–manifolds”, (M Yamasita, editor), volume 23 (2007) 1–14
  • P Kirk, C Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson–Gordon invariants, Topology 38 (1999) 635–661
  • J Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358–426
  • J W Milnor, Infinite cyclic coverings, from: “Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967)”, Prindle, Weber & Schmidt, Boston (1968) 115–133
  • J Porti, Mayberry–Murasugi's formula for links in homology 3–spheres, Proc. Amer. Math. Soc. 132 (2004) 3423–3431
  • J-P Serre, Représentations linéaires des groupes finis, revised edition, Hermann, Paris (1978)
  • D S Silver, S G Williams, Dynamics of twisted Alexander invariants, Topology Appl. 156 (2009) 2795–2811
  • V G Turaev, Reidemeister torsion in knot theory, Uspekhi Mat. Nauk 41 (1986) 97–147, 240 English translation: Russian Math. Surveys 41 (1986) 119–182
  • V Turaev, Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel (2001) Notes taken by Felix Schlenk
  • V Turaev, Torsions of 3–dimensional manifolds, Progress in Mathematics 208, Birkhäuser, Basel (2002)