arXiv research

Satellite links with many twists have simpler companions.

Twisted Alexander invariants have been defined for any knot and linear representation of its group. The invariants are generalized for any periodic representation of the commutator subgroup of the knot group. Properties of the new twisted invariants are given. Under suitable hypotheses, reciprocality and bounds on the …

Study representation varieties of twisted Hopf links using combinatorial and Hodge theory.

We observe the twisted Alexander polynomial for metabelian representations of knot groups into SL(2,C) and study relations to the characterizations of metabelian representations in the character varieties. We give a factorization of the twisted Alexander polynomial for irreducible metabelian representations with the ad…

The Pontryagin dual of the twisted Alexander module for a d-component link and GL(N,Z) representation is an algebraic dynamical system with an elementary description in terms of colorings of a diagram. In the case of a knot, its associated topological entropy is the logarithmic growth rate of the number of torsion elem…

The paper explores mapping class group quotients by Dehn twists and their representations.

The paper calculates Alexander polynomials for knots using finite group representations.

In this paper we will study properties of twisted Alexander polynomials of knots corresponding to metabelian representations. In particular we answer a question of Wada about the twisted Alexander polynomial associated to the tensor product of two representations, and we settle several conjectures of Hirasawa and Muras…

This paper studies twisted signature invariants and twisted linking forms, with a view towards obstructions to knot concordance. Given a knot $K$ and a representation $ρ$ of the knot group, we define a twisted signature function $σ_{K,ρ} \colon S^1 \to \mathbb{Z}$. This invariant satisfies many of the same algebraic pr…

The paper explores representations of specific knot groups and their properties.

Stratifies representation varieties of twisted Hopf links.

Study of knot invariants using twisted Iwasawa theory.

Paper discusses groups where twisted Alexander polynomials vanish.

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 …

Researchers study Prym representations for handlebody groups, focusing on cyclic cases.

We study the asymptotic behavior of the twisted Alexander polynomial for the sequence of SL(n ,C)-representations induced from an irreducible metabelian SL(2, C)-representation of a knot group. We give the limits of the leading coefficients in the asymptotics of the twisted Alexander polynomial and related Reidemeister…

A conjecture of Riley about the relationship between real parabolic representations and signatures of two-bridge knots is verified for double twist knots.

Study subgroup actions on mapping class groups using Heisenberg representations.

Survey on twisted dynamical zeta functions and Fried's conjecture.

Study on balanced Hermitian structures on Lie algebras twisted by representations.

Study calculates twisted Alexander polynomials for Montesinos knots.

We calculate the twisted Reidemeister torsion of the complement of an iterated torus knot associated with a representation of its fundamental group to the complex special linear group of degree two. We also show that the twisted Reidemeister torsions associated with various representations appear in the asymptotic expa…

Let $Γ$ be the fundamental group of the exterior of a knot in the three-sphere. We study deformations of representations of $Γ$ into $\mathrm{SL}_n(\mathbf{C})$ which are the sum of two irreducible representations. For such representations we give a necessary condition, in terms of the twisted Alexander polynomial, for…

Study kernels of mapping class group representations on surface configuration spaces.

We calculate the twisted Alexander polynomials of $(-2,3,2n+1)$-pretzel knots associated to their holonomy representations. As a corollary, we obtain new supporting evidences of Dunfield, Friedl and Jackson's conjecture, that is, the twisted Alexander polynomials of hyperbolic knots associated to their holonomy represe…

Paper uses Long-Moody construction for new braid group representations.

Researchers compute cohomology of mapping class groups with Prym representations, showing instability for large genus.

Study of twisted Ruelle zeta function on hyperbolic manifolds and its relation to analytic torsion.

Given a knot complement X and its p-fold cyclic cover X_p, we identify twisted polynomials associated to 1-dimensional linear representations of the fundamental group of X_p with twisted polynomials associated to related p-dimensional linear representations of the fundamental group of X. This provides a simpler and fas…

Calculates Dehn twist actions on conformal blocks for modular categories.

Explicit matrix presentations of Blanchfield pairings and twisted pairings for torus knots.

The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.

Cataclysm deformations study Anosov representations and their convergence.

The paper studies submonoids of singular twisted virtual braids and their properties.

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 th…

Cataclysm deformations study Anosov representations, leading to new formulas and non-open sets.

We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible SL(2;C)-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We al…

For a complete hyperbolic three manifold M, we consider the representations of its fundamental group obtained by composing a lift of the holonomy with complex finite dimensional representations of SL(2,C). We prove a vanishing result for the cohomology of M with coefficients twisted by these representations, using tech…

We show that any twisted Dijkgraaf-Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof, Rowell and Witherspoon showing that the braid group images are finite. In particular, our result answers their question regarding finitene…

Twisted links are obtained from a base link by starting with a $n$-braid representation, choosing several ($m$) adjacent strands, and applying one or more twists to the set. Various restrictions may be applied, e.g. the twists may be required to be positive or full twists, or the base braid may be required to have a ce…

Ordinarily, quiver varieties are constructed as moduli spaces of quiver representations in the category of vector spaces. It is also natural to consider quiver representations in a richer category, namely that of vector bundles on some complex variety equipped with a fixed sheaf that twists the morphisms. Representatio…

The group of a nontrivial knot admits a finite permutation representation such that the corresponding twisted Alexander polynomial is not a unit.

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…

Study on kernels of SO(3) WRT representations for surfaces of genus g≥3.

New insights into cohomology of closed 1-forms.

We define the twisted Blanchfield pairing of a symmetric triad of chain complexes over a group ring Z[G], together with a unitary representation of G over an Ore domain with involution. We prove that the pairing is sesquilinear, and we prove that it is hermitian and nonsingular under certain extra conditions. A twisted…

We investigate how one can twist L^2-invariants such as L^2-Betti numbers and L^2-torsion with finite-dimensional representations. As a special case we assign to the universal covering of a finite connected CW-complex X together with an element phi in H^1(X;R) a phi-twisted L^2-torsion function from R^{>0} to R, provid…

Study non-acyclic SL2-representations of twist knots and their L-functions.