Reidemeister Torsion Research Articles

Reidemeister torsion of a 3-manifold obtained by an integral Dehn-surgery along the figure-eight knot

Let M be a 3-manifold obtained by a Dehn-surgery along the figure-eight knot. We give a formula of the Reidemeisiter torsion of M for any SL(2; C)-irreducible representation. It has a rational expression of the trace of the image of the meridian.

Twisted Alexander polynomials, character varieties and Reidemeister torsions of double branched covers

We give an extension of Fox's formula of the Alexander polynomial for a double branched cover over the three-sphere. Our formula provides the Reidemeister torsion of the double branched cover along a knot for a non-trivial 1-dimensional representation. In our formula, the Reidemeister torsion is given by the product of two factors derived from the knot group. One of the factors is determined by the twisted Alexander polynomial and the other is determined by a rational function on the character variety of the knot group. As an application, we show that these products distinguish the isotopy classes of two-bridge knots up to their mirror images.

A functorial extension of the abelian Reidemeister torsions of three-manifolds

Let \mathbb F be a field and let G \subset \mathbb F \setminus \{0\} be a multiplicative subgroup. We consider the category \mathsf {Cob}_G of 3 -dimensional cobordisms equipped with a representation of their fundamental group in G , and the category \mathsf {Vect}_{\mathbb F, \pm G} of \mathbb F -linear maps defined up to multiplication by an element of \pm G . Using the elementary theory of Reidemeister torsions, we construct a "Reidemeister functor" from \mathsf {Cob}_G to \mathsf {Vect}_{\mathbb F, \pm G} . In particular, when the group G is free abelian and \mathbb F is the field of fractions of the group ring \mathbb Z [G] , we obtain a functorial formulation of an Alexander-type invariant introduced by Lescop for 3 -manifolds with boundary; when G is trivial, the Reidemeister functor specializes to the TQFT developed by Frohman and Nicas to enclose the Alexander polynomial of knots. The study of the Reidemeister functor is carried out for any multiplicative subgroup G\subset \mathbb F \setminus \{0\} . We obtain a duality result and we show that the resulting projective representation of the monoid of homology cobordisms is equivalent to the Magnus representation combined with the relative Reidemeister torsion.

Connected sum of orientable surfaces and Reidemeister torsion

Connected sum of orientable surfaces and Reidemeister torsion

Reidemeister torsion, peripheral complex and Alexander polynomials of hypersurface complements

Let $f:\CN \rightarrow \C $ be a polynomial, which is transversal (or regular) at infinity. Let $\U=\CN\setminus f^{-1}(0)$ be the corresponding affine hypersurface complement. By using the peripheral complex associated to $f$, we give several estimates for the (infinite cyclic) Alexander polynomials of $\U$ induced by $f$, and we describe the error terms for such estimates. The obtained polynomial identities can be further refined by using the Reidemeister torsion, generalizing a similar formula proved by Cogolludo and Florens in the case of plane curves. We also show that the above-mentioned peripheral complex underlies an algebraic mixed Hodge module. This fact allows us to construct mixed Hodge structures on the Alexander modules of the boundary manifold of $\U$.

Normalization of twisted Alexander invariants

Twisted Alexander invariants of knots are well defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide with sign-determined Reidemeister torsion in a normalized setting, and refine the duality theorem. We further obtain necessary conditions on the invariants for a knot to be fibered, and study behavior of the highest degrees of the invariants.

On a twisted Reidemeister torsion

Given a finite simplicial complex, a unimodular representation of its fundamental group, and a closed twisted cochain of odd degree, we define a twisted version of the Reidemeister torsion, extending a previous definition of V. Mathai and S. Wu. The main tool is a complex of piecewise smooth currents introduced by J. Dupont in 1986.

A GLUING FORMULA FOR THE ANALYTIC TORSION ON HYPERBOLIC MANIFOLDS WITH CUSPS

For an odd-dimensional oriented hyperbolic manifold with cusps and strongly acyclic coefficient systems, we define the Reidemeister torsion of the Borel–Serre compactification of the manifold using bases of cohomology classes defined via Eisenstein series by the method of Harder. In the main result of this paper we relate this combinatorial torsion to the regularized analytic torsion. Together with results on the asymptotic behaviour of the regularized analytic torsion, established previously, this should have applications to study the growth of torsion in the cohomology of arithmetic groups. Our main result is established via a gluing formula, and here our approach is heavily inspired by a recent paper of Lesch.

On the Kashaev invariant and the twisted Reidemeister torsion of two-bridge knots

It is conjectured that, in the asymptotic expansion of the Kashaev invariant of a hyperbolic knot, the first coefficient is represented by the complex volume of the knot complement, and the second coefficient is represented by a constant multiple of the square root of the twisted Reidemeister torsion associated with the holonomy representation of the hyperbolic structure of the knot complement. In particular, this conjecture has been rigorously proved for some simple hyperbolic knots, for which the second coefficient is presented by a modification of the square root of the Hessian of the potential function of the hyperbolic structure of the knot complement. In this paper, we define an invariant of a parametrized knot diagram as a modification of the Hessian of the potential function obtained from the parametrized knot diagram. Further, we show that this invariant is equal (up to sign) to a constant multiple of the twisted Reidemeister torsion for any two-bridge knot. 57M27;

Seifert surgery on knots via Reidemeister torsion and Casson–Walker–Lescop invariant

For a knot K with ΔK(t)≐t2−3t+1 in a homology 3-sphere, let M be the result of 2/q-surgery on K. We show that appropriate assumptions on the Reidemeister torsion and the Casson–Walker–Lescop invariant of the universal abelian covering of M imply q=±1, if M is a Seifert fibered space.

Knot state asymptotics II: Witten conjecture and irreducible representations

This article pursues the study of the knot state asymptotics in the large level limit initiated in Charles and Marché (Knot state asymptotics I. Abelian representations and the A–J conjecture, 2011). As a main result, we prove the Witten asymptotic expansion conjecture for the Dehn fillings of the figure eight knot. The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the $\operatorname {SU}_{2}$ -character manifold of the peripheral torus. In the previous paper, we conjectured that the knot state concentrates on the character variety of the knot with a given asymptotic behavior on the neighborhood of the abelian representations. In the present paper we study the neighborhood of irreducible representations. We conjecture that the knot state is Lagrangian with a phase and a symbol given respectively by the Chern-Simons and Reidemeister torsion invariants. We show that under some mild assumptions, these conjectures imply the Witten conjecture on the asymptotic expansion of WRT invariants of the Dehn fillings of the knot. Using microlocal techniques, we show that the figure eight knot state satisfies our conjecture starting from q-differential relations verified by the colored Jones polynomials. The proof relies on a differential equation satisfied by the Reidemeister torsion along the branches of the character variety, a phenomenon which has not been observed previously as far as we know.

Twisted Reidemeister torsion and the Thurston norm: Graph manifolds and finite representations

We show that the Thurston norm of any irreducible 3-manifold can be detected using twisted Reidemeister torsions corresponding to integral representations and also corresponding to representations over finite fields. In particular our result holds for all graph manifolds, these are not covered by the earlier work of the first author and Vidussi.

Kirchhoff’s theorems in higher dimensions and Reidemeister torsion

Using ideas from algebraic topology and statistical mechanics, we generalize Kirchhoff's network and matrix-tree theorems to finite CW complexes of arbitrary dimension. As an application, we give a formula expressing Reidemeister torsion as an enumeration of higher dimensional spanning trees.

The Nielsen and Reidemeister numbers of maps on infra-solvmanifolds of type [formula omitted

We prove the rationality, the functional equations and calculate the radii of convergence of the Nielsen and the Reidemeister zeta functions of continuous maps on infra-solvmanifolds of type (R). We find a connection between the Reidemeister and Nielsen zeta functions and the Reidemeister torsions of the corresponding mapping tori. We show that if the Reidemeister zeta function is defined for a homeomorphism on an infra-solvmanifold of type (R), then this manifold is an infra-nilmanifold. We also prove that a map on an infra-solvmanifold of type (R) induced by an affine map minimizes the topological entropy in its homotopy class and it has a rational Artin–Mazur zeta function. Finally we prove the Gauss congruences for the Reidemeister and Nielsen numbers of any map on an infra-solvmanifolds of type (R) whenever all the Reidemeister numbers of iterates of the map are finite. Our main technical tool is the averaging formulas for the Lefschetz, the Nielsen and the Reidemeister numbers on infra-solvmanifolds of type (R).

The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern–Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers. If the knot is hyperbolic, the representation can be regarded as a deformation of the holonomy representation that determines the complete hyperbolic structure. In this article, we study a similar phenomenon when the knot is a twice-iterated torus knot. In this case, the asymptotic expansion of the colored Jones polynomial splits into sums, and each summand is related to the Chern–Simons invariant and the Reidemeister torsion associated with a representation.

THE EQUIVARIANT CHEEGER–MÜLLER THEOREM ON LOCALLY SYMMETRIC SPACES

In this paper, we provide a concrete interpretation of equivariant Reidemeister torsion, and demonstrate that Bismut–Zhang’s equivariant Cheeger–Müller theorem simplifies considerably when applied to locally symmetric spaces. In a companion paper, this allows us to extend recent results on torsion cohomology growth and torsion cohomology comparison for arithmetic locally symmetric spaces to an equivariant setting.

Twisted Alexander polynomials with the adjoint action for some classes of knots

We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in [J. Dubois, Nonabelian twisted Reidemeister torsion for fibered knots, Canad. Math. Bull.49(1) (2006) 55–71] and a formula for the nonabelian Reidemeister torsion of twist knots that is better than the one in [J. Dubois, V. Huynh and Y. Yamaguchi, Nonabelian Reidemeister torsion for twist knots, J. Knot Theory Ramifications18(3) (2009) 303–341].

A gluing formula for Reidemeister–Turaev torsion

We extend Turaev's theory of Euler structures and torsion invariants on 3-manifolds to the case of vector fields having generic behavior on the boundary. This allows to easily define gluings of Euler structures and to develop a completely general gluing formula for Reidemeister torsion of 3-manifolds. Lastly, we describe a combinatorial presentation of Euler structures via stream-spines, as a tool to effectively compute torsion.

A note on Reidemeister torsion and pleated surfaces

This paper uses the notion of ℂ-symplectic chain complex and proves an explicit formula for the Reidemeister torsion of an arbitrary ℂ-symplectic chain complex in terms of intersection forms of the homologies. In applications, the formula is applied to closed manifolds and also compact manifolds with boundary by using the homologies with coefficients in complex numbers field. Moreover, an explicit formula for the Reidemeister torsion of representations from the fundamental group of a closed oriented hyperbolic surface to PSL2(ℂ) is presented in terms of the cup product of twisted cohomologies, which is related with Weil–Petersson form and thus the Thurston symplectic form. The formula is also applied to pleated surfaces.

Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps

For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of \(\mathrm{SL }_2(\mathbb {C})\) to the corresponding Reidemeister torsion. While the analytic torsion is a spectral invariant of the manifold, the Reidemeister torsion is of combinatorial nature. Our proof rests on an expression of the analytic torsion in terms of special values of Ruelle zeta functions as well as on recent work of Pere Menal-Ferrer and Joan Porti.