arXiv research

Researchers extend Alexander polynomial to knotoids and linkoids.

By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the i…

Study Alexander polynomials of ribbon and virtual knots using ribbon's intrinsic singularity.

Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots w…

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…

Study Alexander polynomials of special alternating links and generalize Fox's conjecture.

Alexander polynomial equals spanning tree count at t=1.

Proves divisibility relations for symplectic curve polynomials.

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…

Establishes connection between Alexander polynomials and triangulations.

A new invariant for links generalizes Alexander polynomial for sl_3.

Clock theorem extended to knotoids and linkoids.

As a generalization of a fundamental result about the Alexander polynomial of links, we give a description of a Torres condition for the twisted Alexander polynomial of links associated to a unimodular representation.

X.S. Lin's original definition of twisted Alexander knot polynomial is generalized for arbitrary finitely presented groups. J. Cha's fibering obstruction theorem is generalized. The group of a nontrivial virtual knot shown by L. Kauffman to have trivial Jones polynomial is seen also to have a faithful representation th…

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.

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…

For knots in $S^3$, it is well-known that the Alexander polynomial of a ribbon knot factorizes as $f(t)f(t^{-1})$ for some polynomial $f(t)$. By contrast, the Alexander polynomial of a ribbon $2$-knot is not even symmetric in general. Via an alternative notion of ribbon $2$-knots, we give a topological condition on a $…

Paper conjectures Links-Gould invariant generalizes Alexander polynomial.

Paper computes Alexander polynomials for arborescent links.

Study calculates twisted Alexander polynomials for Montesinos knots.

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

Formula for Alexander polynomial of links with twists.

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek poly…

Explicit formulas for pretzel knots' Alexander polynomials.

We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties. For knots with point singularities, we obtain a classification of these polynomial…

New Alexander polynomial for singular knots improves upon existing methods.

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

Paper discusses groups where twisted Alexander polynomials vanish.

Extended symmetric union with multiple tangle regions and Alexander polynomial properties.

In this paper, we study distribution of the zeros of the Alexander polynomials of knots and links in S^3. We call a knot or link "real stable" (resp. "circular stable") if all the zeros of its Alexander polynomial are real (resp. unit complex). We give a general construction of real stable and circular stable knots and…

We show how Seifert surfaces, so useful for the understanding of the Alexander polynomial Δ_L(t), can be generalized in order to study the multivariable Alexander polynomial Δ_L(t_1,...,t_μ). In particular, we give an elementary and geometric proof of the Torres formula.

A simplified proof of the Alexander-Conway polynomial exists.

Homology handles with trivial Alexander polynomial bound a 3D sphere.

The paper studies the n-loop Kontsevich invariant of knots with same Alexander polynomial.

Study Alexander polynomials of links in 3-torus.

New proof of trapezoidal property for Alexander polynomials of special alternating links.

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

We provide the twisted Alexander polynomials of finite abelian covers over three-dimensional manifolds whose boundary is a finite union of tori. This is a generalization of a well-known formula for the usual Alexander polynomial of knots in finite cyclic branched covers over the three-dimensional sphere.

Paper constructs a new approach to extract Affine Index Polynomial from Sawollek Polynomial.

Study connects knot polynomials with number theory sums.

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…

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…

4-move kills Alexander polynomial

Direct proof of Alexander polynomial scaling for L-shaped representations.

New methods compute Alexander polynomials for complex knots.

Proves cosmetic crossing conjecture for certain knots.

We propose an algorithm which allows to derive the generalized Alexander polynomial invariants of knots and links with the help of the q,p-numbers, appearing in bosonic two-parameter quantum algebra. These polynomials turn into HOMFLY ones by applying special parametrization. The Jones polynomials can be also obtained …

Alexander invariant created for doodles, vanishes on unlinked doodles.