arXiv research

Alexander polynomial equals spanning tree count at t=1.

Paper introduces clock moves for plane graphs and proves Alexander polynomial properties.

New proof and formula linking fusion trees to quantum knot invariants.

A new knot invariant using tangle-valued 1-cocycles.

This talk is a report on joint work with A. Vaintrob [arXiv:math.CO/0109104 and math.GT/0111102]. It is organised as follows. We begin by recalling how the classical Matrix-Tree Theorem relates two different expressions for the lowest degree coefficient of the Alexander-Conway polynomial of a link. We then state our fo…

The paper improves bounds on the complexity of computing link polynomials.

Minor typographical errors fixed. Cochran constructed many links with Alexander module that of the unlink and some nonvanishing Milnor invariants, using as input commutators in a free group and as an invariant the longitudes of the links. We present a different and conjecturally complete construction, that uses element…

We study relations between the Alexander-Conway polynomial $\nabla_L$ and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of $\nabla_L$ of an m-component link L all of whose Milnor numbers $μ_{i_1... i_p}$ van…

The paper improves bounds on skein tree depth and delta-crossing numbers for knots and links.

To every tree we associate a filtered cochain complex. Its cohomology and the corresponding spectral sequence have clear combinatorial description. If a tree is the Dynkin diagram of a simple plane curve singularity, the graded Euler characteristic of this complex coincides with the Alexander polynomial of the link. In…

The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix. We prove that in the case of three-graphs (that is, hypergraphs whose edges have exactly three vertices) the spanning trees are generated by the Pfaffian of a suitably…

Extends classical results to virtual links, proving new properties of alternating and semi-alternating virtual links.

We give constructions to realize an odd number, which is representable as sum of two squares, as determinant of an achiral knot, thus proving that these are exactly the numbers occurring as such determinants. Later we study which numbers occur as determinants of prime alternating achiral knots, and obtain a complete re…

Balloons are two-dimensional spheres. Hoops are one dimensional loops. Knotted Balloons and Hoops (KBH) in 4-space behave much like the first and second homotopy groups of a topological space - hoops can be composed as in π_1, balloons as in π_2, and hoops "act" on balloons as π_1 acts on π_2. We observe that ordinary …

Alternating-sign Hopf plumbing along a tree yields fibered alternating links whose homological monodromy is, up to a sign, conjugate to some alternating-sign Coxeter transformation. Exploiting this tie, we obtain results about the location of zeros of the Alexander polynomial of the fibered link complement implying a s…

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.

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.

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 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,…