We introduce L 2 -Alexander torsions for 3-manifolds, which can be viewed as a generalization of the L 2 -Alexander invariant of Li-Zhang. We state the L 2 -Alexander torsions for graph manifolds and we partially compute them for fibered manifolds. We furthermore show that, given any irreducible 3-manifold there exists a coefficient system such that the corresponding L 2 -torsion detects the Thurston norm.

Abstract. In this article, we give an explicit formula to compute the non abelian twisted sign-determined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot.

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For a knot K in $S^3$ and a regular representation $ ho$ of its group $G_K$ into SU(2) we construct a non abelian Reidemeister torsion on the first twisted cohomology group of the knot exterior. This non abelian Reidemeister torsion provides a volume form on the SU(2)-representation space of $G_K$. In another way, we construct according to Casson--or more precisely taking into account Lin's and Heusener's further works--a volume form on the SU(2)-representation space of $G_K$. Next, we compare these two apparently different points of view--the first by means of the Reidemeister torsion and the second defined ``a la Casson"--and finally prove that they define the same topological knot invariant

Abstract. We present an invariant of a three-dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. Our invariant is related to a finite-dimensional fermionic topological quantum field theory.

In the asymptotic expansion of the hyperbolic specification of the colored Jones polynomial of torus knots, we identify different geometric contributions, in particular Chern–Simons invariant and twisted Reidemeister torsion with coefficients in the adjoint representation

This paper gives an explicit formula for the SL 2 (C)−non-abelian Reidemeister torsion as defined in [Dub06] in the case of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot. . , i.e. the word w is obtain from w by changing each of its letters by its reverse. Of course this choice is strictly equivalent to presentation (1). But in a sense, when m > 0 the word w m does not give a "reduced" relation (some cancelations are possible in w m xw −m y −1 ) which is not the case for the word Ω m . Some more elementary properties of twist knots are discussed in the following remark. 3 6 7

Abstract. We show that the L 2 -Alexander torsion of a 3-manifold is symmetric. This can be viewed as a generalization of the symmetry of the Alexander polynomial of a knot.