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.
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.
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.
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…
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.
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.
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.
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…
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 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.
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.
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.
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…
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.
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.
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 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.
The paper calculates Alexander polynomials for knots using finite group representations.
problem Calculating Alexander polynomials for knots using specific group representations.
method Defined twisted Alexander polynomials associated with regular representations of finite groups.
result Several formulas for the twisted Alexander polynomial are provided.
Study connects knot polynomials with number theory sums.
problem Alexander polynomials and Dedekind sums of torus knots.
method No specific method mentioned; connects known concepts.
result Established relationship between knot theory and number theory.
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…
Study Alexander polynomials of special alternating links and generalize Fox's conjecture.
problem Distinguish special alternating links up to isotopy using polynomial invariants.
method Combinatorial and discrete geometric properties of Alexander polynomials of special alternating links.
result Generalized Alexander polynomials of special alternating links can be expressed in terms of volumes of root polytopes of unimodular matrices.
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…
Proves divisibility relations for symplectic curve polynomials.
problem Divisibility relations for symplectic curve polynomials.
method New proofs of divisibility relations for Oka and Alexander polynomials of symplectic curves.
result Proves Libgober's divisibility relations for symplectic curves.
4-move kills Alexander polynomial
problem Whether the 4-move is an unknotting operation
method showing every knot can be reduced via 4-moves and isotopies
result every knot can be reduced to one with a trivial Alexander polynomial
Direct proof of Alexander polynomial scaling for L-shaped representations.
problem Proving scaling property of Alexander polynomials for specific representations.
method Direct use of Reshetikhin-Turaev formalism to compute R-matrices.
result Normalized Alexander polynomial for one-hook representations scales with q∣R∣. 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.
Proves cosmetic crossing conjecture for certain knots.
problem Cosmetic crossing conjecture for specific knot types.
method Proof for knots with non-trivial Alexander polynomial; additional assumptions for trivial Alexander polynomial.
result Proves conjecture for specified knot types.
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 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.
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…
We show how the Alexander polynomial of links in lens spaces is related to the classical Alexander polynomial of a link in the 3-sphere, obtained by cutting out the exceptional lens space fibre. It follows from these relationship that a certain normalization of the Alexander polynomial satisfies a skein relation in len…
The taut polynomial equals a twisted Alexander polynomial.
problem Understanding the relationship between taut polynomials and Alexander polynomials.
method Defined taut polynomial of veering triangulations and proved it equals a twisted Alexander polynomial.
result The taut polynomial equals a twisted Alexander polynomial of the underlying manifold.
We observe that Clay-Rolfsen's obstruction of bi-orderability, which uses the classical Alexander polynomial, is not strengthened by using the twisted Alexander polynomials for finite representations unlike many known applications of the Alexander polynomial. This is shown by studying the maximal ordered abelian quotie…
Clock theorem extended to knotoids and linkoids.
problem Generalizing the Clock Theorem to knotoids and linkoids.
method Extending the Clock Theorem to knotoids and linkoids.
result Clock states of knotoid diagrams form a lattice under transpositions.
Study proves Alexander polynomials of certain 4-braid knots satisfy a conjecture and gives formulas for log-concave sequences.
problem Proving the Alexander polynomials of certain 4-braid knots satisfy Fox's Trapezoidal Conjecture.
method Analyzes families of alternating 4-braids and n-braids, providing explicit formulas and verifying log-concavity. result Explicit formulas for signature and first 4 coefficients of Alexander polynomials, showing log-concavity.
Character variety of Borromean link solved, Alexander polynomial formula found.
problem Character variety of Borromean link
method Determined irreducible SL(2,C) character variety, found Alexander polynomial formula
result Formula for twisted Alexander polynomial on character variety
This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice knots.
We give a characterization for the Alexander Polynomials of closed orientable 3-manifolds M with first Betti number 1, as well as some partial results for the characterization problem for M having first Betti number > 1. We first prove an analogue of a theorem of Levine: that the product of an Alexander polynomial of M…
We show that if a link J in the 3-sphere is homotopy ribbon concordant to a link L then the Alexander polynomial of L divides the Alexander polynomial of J.
New Alexander polynomial defined for transverse graphs.
problem Defining a polynomial invariant for transverse graphs.
method Rotation number extension and multi-variable Alexander polynomial.
result Invariant coincides with Uq(gl(1∣1))-Alexander polynomial. Study revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links.
problem Understanding polynomial invariants of pretzel links.
method Revisits Alexander-Conway and Kauffman bracket polynomials for pretzel links P(1,1,n). result Reveals properties of Alexander-Conway and Kauffman bracket polynomials for P(1,1,n). 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.
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.
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.
A new invariant for links generalizes Alexander polynomial for sl_3.
problem Defining a non-abelian generalization of the Alexander polynomial.
method Using quantum sl3 representations and Laurent polynomials. result Established a direct relation between Δsl3 and the Alexander polynomial. 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.
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…