Alexander method extended to infinite-type surfaces.
problem Determining equality of elements in mapping class groups.
method Combinatorial tool extended to infinite-type surfaces.
result Verification of relations and triviality of centers in mapping class groups.
Paper computes Alexander polynomials for arborescent links.
problem Explicit formulas for Alexander polynomials are hard to compute for most link families.
method Efficient method for arborescent links, using recursive polynomials.
result Explicit closed formulas for pretzel links derived.
New methods compute Alexander polynomials for complex knots.
problem Efficiently computing higher order Alexander polynomials for complex knots.
method Developed new algorithms to compute the Smith normal form of Alexander matrices.
result Computed Alexander polynomials for knots up to 100 crossings.
New Alexander polynomial for singular knots improves upon existing methods.
problem Defining a polynomial invariant for singular knots.
method Introducing a perturbed Alexander polynomial.
result The new polynomial agrees with previous definitions for long knots.
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…
This paper deals with the study of a new family of knot invariants: the L2-Alexander invariant. A main result is to give a method of computation of the L2-Alexander invariant of a knot complement using any presentation of default 1 of the knot group.
Paper proves almost all stabilizer subgroups of Thompson's group satisfy Alexander's theorem.
problem Alexander's theorem for stabilizer subgroups of Thompson's group.
method Defined a method to construct knots and links from Thompson's group F and proved Alexander's theorem for stabilizer subgroups.
result Almost all stabilizer subgroups under the natural action on the unit interval satisfy Alexander's theorem.
Study of twisted Alexander matrices for certain quandles and their invariants.
problem Investigate f-twisted Alexander matrices for quandles associated with Alexander pairs. method Define and analyze f-twisted Alexander matrices of certain quandles, relate to Carter-Saito-Satoh's invariant, and discuss connections to quandle homology groups. result 0-th elementary ideal of f-twisted Alexander matrix can be described using Carter-Saito-Satoh's invariant. Establishes connection between Alexander polynomials and triangulations.
problem Alexander polynomials and their variants for knots.
method Introduces twisted Neumann--Zagier matrices for ideal triangulations.
result Formulas for Alexander polynomial and its variants.
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…
New method computes knot invariants using free group automorphisms.
problem Computing knot invariants efficiently and accurately.
method Using representations of braid groups by automorphisms of a free group.
result Compared isotopic invariants to Alexander polynomials.
Researchers extend Alexander polynomial to knotoids and linkoids.
problem Defining and studying Alexander polynomial extensions for knotoids and linkoids.
method Developed and proved conjecture on mock Alexander polynomial for knotoids and linkoids.
result Proved conjecture on mock Alexander polynomial for knotoids and linkoids.
The paper reinterprets knot group invariants using affine transformations.
problem Alexander invariants of knots and their geometric interpretation.
method Representation varieties of knot groups into extrmAGL1(C). result Alexander polynomial as the singular locus of a coherent sheaf.
Study Alexander matrices for link quandles and their relation to knot invariants.
problem Understanding Alexander matrices for link quandles and their applications to knot invariants.
method Investigate f-twisted Alexander matrices and their connection to quandle cocycle invariants. result Show that f-twisted Alexander invariants of knot quandles are stronger than those of knot groups. Study Alexander polynomials of ribbon and virtual knots using ribbon's intrinsic singularity.
problem Determining Alexander polynomials for ribbon and virtual knots.
method Using ribbon's intrinsic singularity information, defining half Alexander polynomial, and developing simplified formulas.
result New formulas for Alexander polynomials of general knots and virtual knots in terms of Gauss diagrams.
Paper discusses groups where twisted Alexander polynomials vanish.
problem Understanding groups with vanishing twisted Alexander polynomials.
method Introduced and studied twisted Alexander vanishing groups, constructed knots, and analyzed representations.
result Every faithful irreducible representation of a TAV group causes the twisted Alexander polynomial to be zero.
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…
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.
problem Tackles the calculation of twisted Alexander polynomials for Montesinos knots.
method Uses SL2(C)-representations to calculate leading coefficients and degrees of the polynomials. result Obtains non-monic twisted Alexander polynomials for some nonfibered knots.
The paper studies twisted Alexander polynomials for knot groups in various extensions.
problem Understanding twisted Alexander polynomials in knot groups for different extensions.
method Developed mod p formula for twisted Alexander polynomials and studied central extensions.
result Established formulas for twisted Alexander polynomials in knot groups for various extensions.
A new, simple method to compute a knot invariant.
problem Computing a powerful knot invariant efficiently.
method Using a quadratic expression in the entries of the inverse of a standard matrix.
result The method leads to a powerful knot invariant with concise formulas.
Formula for Alexander polynomial of links with twists.
problem Computing Alexander polynomial of links with twists.
method Using vector space representation of Uq(gl(1∣1)). result Alexander polynomials stabilize after adding enough twists.
Study on a knot invariant's vanishing order.
problem Understanding the vanishing order of twisted Alexander polynomials.
method Defined and explored properties of the twisted Alexander vanishing order.
result Listed twisted Alexander vanishing groups of order less than 201.
Study on periodic knots, proving limitations on their Alexander polynomials.
problem Understanding Alexander polynomials of periodic knots.
method Polynomial factorization, number theory interpretation, computational methods.
result Alexander polynomials of freely periodic knots are restricted to products of cyclotomic polynomials.
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.
problem Existence of the Alexander-Conway polynomial for links in 3D space.
method Presented an accurate detailed exposition of the proof.
result Existence of the Alexander-Conway polynomial proved.
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 L2-Alexander invariants of knots. We quickly recall the definitions and we summarize an…
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…
A new method computes link invariants from diagrams.
problem Computing link invariants efficiently.
method Single symmetric matrix from a link diagram.
result Multivariable Alexander polynomial computation.
Paper uses Long-Moody construction for new braid group representations.
problem Constructing new representations of braid groups.
method Applies Fox derivation to matrix presentation of Long-Moody construction.
result Shows relation to twisted Alexander invariants.
New Alexander invariants for knot groups computed using K1-groups.
problem Computing Alexander invariants for knot groups.
method Introducing K1-classes and comparing them with other Alexander polynomials. result Non-triviality of computed K1-classes for some knots. In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial ΔL(t) is vanishing, then L admits a non-trivial coloring by any non-trivial Alexander quandle Q, and that if ΔL(t)=1, then L admits only the trivial coloring by any Alexa…
Given a virtual knot K, we construct a group VGK called the virtual knot group, and we use the elementary ideals of VGK to define invariants of K called the virtual Alexander invariants. For instance, associated to the k=0 ideal is a polynomial HK(s,t,q) in three variables which we call the virtual Alexa…
Alexander quandles can be embedded into groups.
problem Embedding Alexander quandles into groups.
method For any twisted conjugate quandle, find a group such that the quandle is embedded into the conjugation quandle of the group.
result Alexander quandles can be embedded into groups.
Explicit formulas for pretzel knots' Alexander polynomials.
problem Alexander polynomial of pretzel knots
method Provided explicit formulas
result Characterization of pretzel knots with trivial Alexander polynomial
New examples of Cappell-Shaneson knot pairs with same Alexander polynomial found.
problem Determine dimensions for non-reflexive knot pairs.
method Constructing new examples of Cappell-Shaneson knot pairs.
result Found examples of Cappell-Shaneson knot pairs with same Alexander polynomial but inequivalent.
We derive a factorization of the Alexander polynomial of the 4-strand Turk's head knot using hypergeometric representations.
problem Deriving a factorization of the Alexander polynomial of the 4-strand Turk's head knot
method Using the reduced Burau representation and multivariable resultant elimination over reciprocal constraints
result Deriving a factorization of the Alexander polynomial in terms of Chebyshev 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.
problem Understanding when homology handles bound 3D spheres.
method Using Freedman and Quinn's result for Z-homology 3-spheres. result A distinguished homology handle with trivial Alexander polynomial bounds a homology S1imesD3. Study Alexander polynomials of links in 3-torus.
problem Investigate Alexander polynomials of links in 3-torus.
method Diagrammatic approach, Reidemeister moves, fundamental group, homology group, Alexander polynomials, twisted Alexander polynomials.
result Computed Alexander and twisted Alexander polynomials of links in 3-torus.
Alexander quandles fail to distinguish certain links, thus not detecting causality.
problem Detecting causality in spacetimes using link polynomials.
method Examined Alexander quandles' ability to distinguish specific links.
result Alexander quandles cannot distinguish the connected sum of two Hopf links and Allen-Swenberg Links.
The study eliminates infinite families of knots with nontrivial Alexander polynomials and improves unknotting number data.
problem Identifying knots with nontrivial Alexander polynomials and improving knot classification.
method Elimination of infinite families of knots and use of determinants to improve unknotting number data.
result Elimination of infinite families of knots with nontrivial Alexander polynomials and improvement of unknotting number data.
Alexander polynomial equals spanning tree count at t=1.
problem Alexander polynomial for spatial graphs.
method Combinatorial constructions generalized to weighted graphs.
result Value of Alexander polynomial at t=1 equals weighted spanning tree count.
Alexander invariant created for doodles, vanishes on unlinked doodles.
problem Creating an Alexander type invariant for doodles.
method Deformation of Tits representation and Chebyshev polynomials of second kind.
result Invariant vanishes on unlinked doodles with more than one component.
Alexander's conjecture extended to infinite simplicial complexes.
problem Alexander's conjecture for infinite simplicial complexes.
method Generalization of recent result for finite simplicial complexes.
result Alexander's conjecture holds for 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.
problem Peripheral elements in reduced Alexander modules
method Analyzes and corrects a paper on Alexander modules
result Answers a question and corrects a minor error in the original paper