arXiv research

Study calculates $L^2$-Alexander torsion for specific geometric spaces.

Formulae for $L^2$-Alexander torsions of links and knots.

New $L^2$-Burau maps compute Alexander torsions for links.

We introduce $L^2$-Alexander torsions for 3-manifolds, which can be viewed as a generalization of the $L^2$-Alexander polynomial 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 ex…

Formulas for knot polynomials and torsions of specific knots.

Study bounds leading coefficients of $L^2$-Alexander torsions for 3-manifolds.

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.

Paper proves a generalized Torres formula for twisted Reidemeister torsion.

Establishes connection between Alexander polynomials and triangulations.

Guts determine the leading coefficients of $L^2$-Alexander torsions for 3-manifolds.

We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We recover the classical torsionness and divisibility results, which say that, unde…

We study a computational method of the hyperbolic Reidemeister torsion (also called in the literature the non-abelian Reidemeister torsion) induced by J. Porti for complete hyperbolic three-dimensional manifolds with cusps. The derivative of the twisted Alexander invariant for a hyperbolic knot exterior gives the hyper…

The Alexander polynomial of a knot has been generalized in three different ways to give twisted invariants. The resulting invariants are usually referred to as twisted Alexander polynomials, higher-order Alexander polynomials and $L^2$-Alexander invariants of knots. We quickly recall the definitions and we summarize an…

New Alexander invariant classes computed for knot group representations.

The paper introduces new invariants for genus one knots and surfaces.

The paper explores algebraic properties of Alexander polynomials and Reidemeister torsions for torus knots.

As a generalization of a classical result on the Alexander polynomial for fibered knots, we show in this paper that the Reidemeister torsion associated to a certain representation detects fiberedness of knots in the three sphere.

We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.

Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister t…

The paper reformulates an invariant and calculates it for lens spaces.

We prove duality theorems for twisted Reidemeister torsions and twisted Alexander polynomials generalizing the results of Turaev. As a corollary we determine the parity of the degrees of twisted Alexander polynomials of 3-manifolds in many cases.

Recently twisted and higher order Alexander polynomials were used by Cochran, Harvey, Friedl--Kim and Turaev to give lower bounds on the Thurston norm. We first show how Reidemeister torsion relates to these Alexander polynomials. We then give lower bounds on the Thurston norm in terms of the Reidemeister torsion which…

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 \cite{Du} and a formula for the nonabelian Reidemeister torsion of twist knots that is better than…

New link invariants derived from $L^2$-Burau maps of braids.

The article computes the Blanchfield pairing for any link.

Formula for twisted Alexander polynomials of torus links.

We give an extension of Fox's formula of the Alexander polynomial for double branched covers over the three-sphere. Our formula provides the Reidemeister torsion of a double branched cover along a knot for a non-trivial one dimensional representation by the product of two factors derived from the knot group. One of the…

Study on higher dimensional Alexander polynomials for metabelian representations of knots.

Study Alexander polynomials to detect fibering and genus of hyperbolic links.

The paper studies norms and invariants for free-by-cyclic groups.

Defines Reidemeister torsion form on 3-manifold character varieties.

We give a short introduction to the theory of twisted Alexander polynomials of a 3--manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications of twisted invarian…

Instanton homology detects 2-torsion in fibered knots.

Proves a theorem on Alexander polynomials for 3-manifolds.

For an irreducible orientable compact $3$-manifold $N$ with empty or incompressible toral boundary, the full $L^2$--Alexander torsion $τ^{(2)}(N,φ)(t)$ associated to any real first cohomology class $φ$ of $N$ is represented by a function of a positive real variable $t$. The paper shows that $τ^{(2)}(N,φ)$ is continuous…

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, an…

Study of Alexander modules for hyperplane arrangements, distinguishing complements.

Formula connects knot invariants to Alexander polynomials.

We generalize a theorem of Bismut-Zhang, which extends the Cheeger-Mueller theorem on Ray-Singer torsion and Reidemeister torsion, to the case where the flat vector bundle over a closed manifold carries a nondegenerate symmetric bilinear form. As a consequence, we prove the Burghelea-Haller conjecture which gives an an…

We compute the characteristic varieties and the Alexander polynomial of a finitely generated nilpotent group. We show that the first characteristic variety may be used to detect nilpotence. We use the Alexander polynomial to deduce that the only torsion-free, finitely generated nilpotent groups with positive deficiency…

The paper examines Seifert surgeries on knots via torsion and invariant.

Study $p$-torsion growth in covers of 3-manifolds, proving Iwasawa formulas.

We study the growth of the order of torsion subgroups of the homology in a tower of finite abelian coverings. In particular, we prove that it is exponential for when the tower converges to the maximal free abelian cover of a link complement when the first nonzero Alexander polynomial has positive logarithmic Mahler mea…

The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4…

Complexity of signed graphs linked to Alexander polynomials and Lehmer's question.

This paper proves bi-orderability of two-bridge link groups.

New nonacyclic Reidemeister torsions defined for odd-dimensional manifolds.

Given a knot and an SL(n,C) representation of its group that is conjugate to its dual, the representation that replaces each matrix with its inverse-transpose, the associated twisted Reidemeister torsion is reciprocal. An example is given of a knot group and SL(3,Z) representation that is not conjugate to its dual for …