arXiv research

Paper analyzes double twist knots using adjoint hyperbolic torsion polynomial.

We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.

Researchers compute twisted Reidemeister torsion for hyperbolic 3-manifolds.

New formula for torsion function in 3-manifolds with torus boundaries.

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…

This is a survey on Reidemeister torsion for hyperbolic three-manifolds of finite volume. Torsions are viewed as topological invariants and also as functions on the variety of representations in $\operatorname{ SL}_2(\mathbb C)$. In both cases, the torsions may also be computed after composing with finite dimensional r…

The paper supports a conjecture about a vanishing identity for certain 3-manifolds.

We establish a link between the holomorphic derivatives of Thurston's hyperbolic gluing equations on an ideally triangulated finite volume hyperbolic 3-manifold and the cohomology of the sheaf of infinitesimal isometries. Moreover, we provide a geometric reformulation of the non-abelian Reidemeister torsion correspondi…

The paper proves a vanishing identity for twist knots using character varieties.

We introduce a vanishing property of adjoint Reidemeister torsions of a cusped hyperbolic 3-manifold derived from the physics of wrapped M5-branes on the manifold. To support our physical observation, we present a rigorous proof for the figure-eight knot complement with respect to all slopes. We also present numerical …

Proves conjecture about integer sums of torus knot torsions.

Researchers compute and predict knot volumes using colored Jones polynomials.

1-loop invariant equals torsion for 2-bridge knots.

New invariants from quantum group theory for hyperbolic 3-manifolds.

The paper analyzes the asymptotic behavior of a knot polynomial for a specific real number.

Explicit formula for Reidemeister torsion of two-bridge knots.

Prove integrality of genus-$g$ indices with adjoint Reidemeister torsions for twist knots and meridians.

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…

New polynomial connects knot genus to 3-manifold geometry.

The colored Jones polynomial of the figure-eight knot connects to an SL(2;R) representation.

Proves freely 2-periodic knots have two canonical components in their character variety.

In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic knots to hyperbolic links, and confirm it for an infinite family of hyperbolic 2…

The paper defines and analyzes the adjoint Reidemeister torsion for connected sums of knots.

New definition of twisted 1-loop invariant using Ptolemy coordinates.

Establishes connection between Alexander polynomials and triangulations.

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.…

New knot theory module shows torsion-ness in number theory.

Study calculates Reidemeister torsions for 3-manifolds from specific surgeries.

In this paper we define the adjoint Reidemeister torsion as a differential form on the character variety of a compact oriented 3-manifold with toral boundary, and prove it defines a regular volume form. Then we show that the torsion form can vanish only at singular points of the character variety. In fact, if the singu…

Paper shows unique decomposition of 3-manifolds and multiplicative property of Reidemeister torsion.

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 invaraint and Reidemeister torsion.

The study connects conic connections and torsion-free principal connections on G-structures.

Homology growth of specific mapping tori vanishes for certain groups.

By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in p…

Derives adjoint polynomials of torus knots in explicit form.

Novel mathematical approach using resurgent analysis reveals new structures in complex Chern-Simons theory.

We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, and give their ex…

We study when the Thurston norm is detected by twisted Alexander polynomials associated to representations of the 3-manifold group to SL(2, C). Specifically, we show that the hyperbolic torsion polynomial determines the genus for a large class of hyperbolic knots in the 3-sphere which includes all special arborescent k…

T. Saito and M. Teragaito asked whether Berge knots of type VII are hyperbolic, and showed that some infinite sequences of the knots are hyperbolic. We show that Berge knots of types VII and VIII are hyperbolic except the known sequence of torus knots. We used the Reidemeister torsions. As a result, the Alexander polyn…

We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a $\mathbb{Z}$-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley…

In this paper, we prove that the Reidemeister torsion twisted by the adjoint representation, which is considered as a 1-form, on the SU(2)-character variety of a knot exterior is invariant under mutation along a Conway sphere.

Geometric derivation of quantum dynamics from Lie group actions.

Semisimple (co)adjoint orbits through real hyperbolic elements are well-known to be symplectomorphic to cotangent bundles. We provide a new proof of this fact based on elementary results on both Lie theory and symplectic geometry. Our proof establishes a new connection between the Iwasawa horospherical projection and t…

New hyperbolic manifolds show exponential homology torsion growth.

Study shows infinite hyperbolic knots with odd torsion in Khovanov homology.

Defines Whitehead torsion for topological spaces via K-theory.

Study shows torsion homology growth vanishes for certain free-by-cyclic groups.

If $M$ is a finite volume complete hyperbolic 3-manifold with one cusp and no 2-torsion, the geometric component $X_M$ of its $\SL(2,\BC)$-character variety is an affine complex curve, which is smooth at the discrete faithful representation $ρ_0$. Porti defined a non-abelian Reidemeister torsion in a neighborhood of $ρ…