arXiv research

In the paper we prove the conjecture by Alexander Zupan that $w(K) \geqslant n^2w(J)$ where w denote the width and $K$ and $J$ are satellite knot and its companion with winding number $n$. Also we proved that for satellite knot with braid pattern, the equality holds.

The paper calculates ribbon numbers for 12-crossing knots using Alexander polynomials.

We generalize bridge trisections to surfaces in four-manifolds, linking them to braided links.

New invariant measures knotted surfaces in 4D, revealing unknottedness.

New method finds infinitely many surface knots with specific bridge numbers.

Meier and Zupan showed that every surface in the four-sphere admits a bridge trisection and can therefore be represented by three simple tangles. This raises the possibility of applying methods from link homology to knotted surfaces. We use link homology to construct an invariant of knotted surfaces (up to isotopy) whi…

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

Establishes connection between Alexander polynomials and triangulations.

Researchers extend Alexander polynomial to knotoids and linkoids.

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

Confirming the Powell Conjecture for genus-3 Heegaard splittings of the 3-sphere.

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

Paper discusses groups where twisted Alexander polynomials vanish.

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 calculates twisted Alexander polynomials for Montesinos knots.

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

Study of 3-manifolds in 5-sphere using bridge decompositions.

Formula for Alexander polynomial of links with twists.

Study on a knot invariant's vanishing order.

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…

A simplified proof of the Alexander-Conway polynomial exists.

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…

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 polynomial for singular knots improves upon existing methods.

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

In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial $Δ_{L}(t)$ is vanishing, then $L$ admits a non-trivial coloring by any non-trivial Alexander quandle $Q$, and that if $Δ_{L}(t)=1$, then $L$ admits only the trivial coloring by any Alexa…

New methods compute Alexander polynomials for complex knots.

Paper computes Alexander polynomials for arborescent links.

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…

Alexander quandles can be embedded into groups.

Explicit formulas for pretzel knots' Alexander polynomials.

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…

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

Study Alexander polynomials of links in 3-torus.

Alexander quandles fail to distinguish certain links, thus not detecting causality.

The study eliminates infinite families of knots with nontrivial Alexander polynomials and improves unknotting number data.

Alexander polynomial equals spanning tree count at t=1.

Alexander invariant created for doodles, vanishes on unlinked doodles.

Alexander's conjecture extended to infinite simplicial complexes.

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. …

Corrects a paper on Alexander modules and answers a related question.

Cochran defined the nth-order integral Alexander module of a knot in the three sphere as the first homology group of the knot's (n+1)th-iterated abelian cover. The case n=0 gives the classical Alexander module (and polynomial). After a localization, one can get a finitely presented module over a principal ideal domain,…

J. Davis showed that the topological concordance class of a link in the 3-sphere is uniquely determined by its Alexander polynomial for 2-component links with Alexander polynomial one. A similar result for knots with Alexander polynomial one was shown earlier by M. Freedman. We prove that these two cases are the only e…

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 this paper, we study surfaces embedded in $4$-manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary $4$-manifold. This extends work of Swenton and Kearton-Kurlin in $S^4$. As an application, we show that bridge trisections of isotopic surfaces in a trisected …

In this article, we present some of the properties of the $L^2$-Alexander invariant of a knot defined by Li and Zhang, some of which are similar to those of the classical Alexander polynomial. Notably we prove that the $L^2$-Alexander invariant detects the trivial knot.

The paper defines conditions for good involutions in generalized Alexander quandles.

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…