arXiv research

Study Alexander matrices for link quandles and their relation to knot invariants.

Study of twisted Alexander matrices for certain quandles and their invariants.

Study on a knot invariant's vanishing order.

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…

We introduce a new algebraic topological technique to detect non-fibred knots in the three sphere using the twisted Alexander invariants. As an application, we show that for any Seifert matrix of a knot with a nontrivial Alexander polynomial, there exist infinitely many non-fibered knots with the given Seifert matrix. …

We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.

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 …

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…

It follows from earlier work of Silver-Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexand…

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…

Analogous zeta function for twisted Alexander invariants defined.

There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handl…

The paper introduces new invariants to refine Alexander polynomials and bounds BNSR Σ-invariants.

New Alexander invariants for knot groups computed using $K_1$-groups.

Let L be an oriented (d+1)-component link in the 3-sphere, and let L(q) be the d-component link in a homology 3-sphere that results from performing 1/q-surgery on the last component. Results about the Alexander polynomial and twisted Alexander polynomials of L(q) corresponding to finite-image representations are obtain…

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

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

Given a virtual knot $K$, we construct a group $VG_K$ called the virtual knot group, and we use the elementary ideals of $VG_K$ to define invariants of $K$ called the virtual Alexander invariants. For instance, associated to the $k=0$ ideal is a polynomial $H_K(s,t,q)$ in three variables which we call the virtual Alexa…

Defines a new knot invariant and studies its properties.

New invariant measures how many twists are needed to unknot welded knots.

Study of knot invariant growth for twisted knots.

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…

In this paper we present a sequence of link invariants, defined from twisted Alexander polynomials, and discuss their effectiveness in distinguish knots. In particular, we recast and extend by geometric means a recent result of Silver and Williams on the nontriviality of twisted Alexander polynomials for nontrivial kno…

New method distinguishes knots and knotted surfaces.

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 quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to gr…

We study how the genus, the simplicial volume and the $L^2$-Alexander invariant of W. Li and W. Zhang can detect individual knots among all others. In particular, we use various techniques coming from hyperbolic geometry and topology to prove that the $L^2$-Alexander invariant contains strictly more information than th…

New Alexander invariant classes computed for knot group representations.

Establishes connection between Alexander polynomials and triangulations.

Paper discusses groups where twisted Alexander polynomials vanish.

The paper studies twisted Alexander polynomials for knot groups in various extensions.

Formula for Alexander polynomial of links with twists.

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…

The theory of quandle (co)homology and cocycle knot invariants is rapidly being developed. We begin with a summary of these recent advances. One such advance is the notion of a dynamical cocycle. We show how dynamical cocycles can be used to color knotted surfaces that are obtained from classical knots by twist-spinnin…

Study calculates twisted Alexander polynomials for Montesinos knots.

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…

We study torsion properties of the twisted Alexander modules of the affine complement $M$ of a complex essential hyperplane arrangement, as well as those of punctured stratified tubular neighborhoods of complex essential hyperplane arrangements. We investigate divisibility properties between the twisted Alexander polyn…

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.

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…

Classifies negative-twisting structures on Seifert fibred spaces using Heegaard Floer homology.

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

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.

Let $Δ_{L,ρ_n}(t)$ be the twisted Alexander polynomial with respect to the representation given by the composition of the lift of the holonomy representation of a certain hyperbolic link $L$ and the $n$-dimensional irreducible complex representation of $\text{SL}(2,\mathbb C)$. We consider a sequence of $Δ_{L,ρ_n}(t)$ …

Formula for Alexander polynomial of twisted torus knots derived.

In recent years, twisted Alexander polynomial has been playing an important role in low-dimensional topology. For Montesinos links, we develop an efficient method to compute the twisted Alexander polynomial associated to any linear representation. In particular, formulas for multi-variable Alexander polynomials of thes…

Classifies small links in an unmarked solid torus.

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

If phi: G-->G' is a surjective homomorphism, we prove that the twisted Alexander polynomial of G is divisible by the twisted Alexander polynomial of G'. As an application, we show non-existence of surjective homomorphism between certain knot groups.