A new property of the Alexander polynomial

卫星链环的高阶Alexander多项式

Higher Alexander Polynomials of Satellite Links

本文提出了卫星链环的高阶Alexander多项式的计算公式。

On the twisted Alexander polynomial for metabelian representations into SL(2, C)

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 adjoint action on sl(2,C), in which the Alexander polynomial and the twisted Alexander polynomial appear as factors. We also show several explicit examples.

Burau maps and twisted Alexander polynomials

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.

Twisted Alexander polynomials of periodic knots

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 with 11 crossings. Hartley found a restrictive condition satisfied by the Alexander polynomial of any freely periodic knot. We generalize this result to the twisted Alexander polynomial and illustrate the applicability of this extension in cases in which Hartley's criterion does n…

Adjoint twisted Alexander polynomials of genus one two-bridge knots

We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.

Homology handles with trivial Alexander polynomial

Using Freedman and Quinn's result for $\mathbb{Z}$-homology 3-spheres, we show that a 3-dimensional homology handle with trivial Alexander polynomial bounds a homology $S^1\times D^3$. As a consequence, a distinguished homology handle with trivial Alexander polynomial is topologically null $\widetilde{H}$-cobordant.

Cosmetic crossing conjecture for genus one knots with non-trivial Alexander polynomial

We prove the cosmetic crossing conjecture for genus one knots with non-trivial Alexander polynomial. We also prove the conjecture for genus one knots with trivial Alexander polynomial, under some additional assumptions.

A Fox-Milnor theorem for the Alexander polynomial of knotted $2$-spheres in $S^4$

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 $2$-knot that implies the factorization of the Alexander polynomial.

On the Alexander polynomial of links in lens spaces

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 lens spaces.

The Weight System of the Multivariable Alexander Polynomial

We derive a formula for the weight system of the multivariable Alexander polynomial using determinants, show that it obeys known relations, and satisfies some of the same relations as the single variable polynomial.

Introduction to twisted Alexander polynomials and related topics

This article is based on the lectures in the Winter Braids V (Pau, Feb. 2015). Main puposel of this is to explain how to compute twisted Alexander polynomials for non-experts.

Alexander polynomials and signatures of some high-dimensional knots

We give necessary and sufficient conditions for an integer to be the signature of a (4q-1)-knot in the (4q+1)-sphere with a given square-free Alexander polynomial.

Alexander-Lin twisted polynomials

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 that yields a trivial twisted Alexander polynomial.

Homotopy ribbon concordance and Alexander polynomials

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.

Twisted Alexander polynomials of torus links

In this paper we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the $SL(2, \mathbb C)$-character variety. We also discuss similar things for the higher dimensional twisted Alexander polynomial and the Reidemeister torsion.

The Alexander polynomial at prime powers

When $K$ is a knot and $p \gg 0$ is a prime, we discuss a finite set whose cardinality is $Δ_K(p^n)$, the value of the Alexander polynomial of $K$ at $p^n$.

Instanton homology and the Alexander polynomial

We prove that the instanton knot homology KHI(K) as defined by Kronheimer and Mrowka (Knots, sutures and excision, preprint), recovers the Alexander polynomial for knots K in the 3-sphere.

Alexander polynomial and spanning trees

Inspired by the combinatorial constructions in earlier work of the authors that generalized the classical Alexander polynomial to a large class of spatial graphs with a balanced weight on edges, we show that the value of the Alexander polynomial evaluated at $t=1$ gives the weighted number of the spanning trees of the graph.

Various stabilities of the Alexander polynomials of knots and links

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 links. We also study pairs of real stable knots and links such that the zeros of the Alexander polynomials are interlaced.

Twisted Alexander Polynomials of $(-2,3,2n+1)$-Pretzel Knots

We calculate the twisted Alexander polynomials of $(-2,3,2n+1)$-pretzel knots associated to their holonomy representations. As a corollary, we obtain new supporting evidences of Dunfield, Friedl and Jackson's conjecture, that is, the twisted Alexander polynomials of hyperbolic knots associated to their holonomy representations determine the genus and fiberedness of the knots.

具有Alexander多项式1和高解结数的光滑切片纽结

Smoothly slice knots with Alexander polynomial 1 and high unknotting number

本文证明存在一个同时具有光滑双切片性、双向手性、Alexander多项式为1且解结数为5的纽结，揭示了这些拓扑性质可以共存。

The Alexander polynomial for closed braids in lens spaces

We present a reduced Burau-like representation for the mixed braid group on one strand representing links in lens spaces and show how to calculate the Alexander polynomial of a link directly from the mixed braid.

计算树状链环的亚历山大多项式

Computing Alexander polynomials for arborescent links

本文提出了一种高效计算树状链环亚历山大多项式的方法，并将其应用于蒙泰西诺斯链环和椒盐卷饼链环，得到了显式闭公式。

A Note on Alexander Polynomials of 2-Bridge Links

A formula for the Alexander polynomial of a 2-bridge knot or link given by Hartley and also by Minkus has a beautiful interpretation as a walk on the integers. We extend this to the 2-variable Alexander polynomial of a 2-bridge link, obtaining a formula that corresponds to a walk on the 2-dimensional integer lattice.

Twisted Alexander polynomials of 2-bridge knots

We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.

Using Twisted Alexander Polynomials to Detect Fiberdness

In this paper we use twisted Alexander polynomials to prove that the exterior of a particular graph knot is not fibered. Then we build three 2-component graph links out of this knot, and use similar techniques to discuss their fiberedness.

Studying the multivariable Alexander polynomial by means of Seifert surfaces

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 restriction on the Alexander polynomials of $L$-space knots

Using an invariant defined by Rasmussen, we extend an argument given by Hedden and Watson which further restricts which Alexander polynomials can be realized by $L$-space knots.

奇异纽结的扰动 Alexander 多项式

A perturbed Alexander polynomial for singular knots

本文引入奇异纽结与tangle的Alexander多项式版本，通过引入扰动显著增强其功能，并证明对于奇异长结该多项式与先前定义一致。

链环的符号差跳跃与 Alexander 多项式

Signature Jumps and Alexander Polynomials for Links

本文建立链环符号差函数跳跃点与首个非零高阶 Alexander 多项式根之间的对应关系。

On zeros of the Alexander polynomial of an alternating knot

We prove that for any zero α of the Alexander polynomial of a two-bridge knot, -3 < Re(α) < 6. Furthermore, for a large class of two-bridge knots we prove -1<Re(α).

On the Characterization Problem of Alexander Polynomials of Closed 3-Manifolds

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 with a symmetric polynomial in the same number of variables having non 0 trace, is again an Alexander polynomial of a closed orientable 3-manifold. Using the fact that there exists M with Alexander polynomial = 1 for M with first Betti number 1, 2 or 3, we conclude that symmetri…

Metabelian SL(n,C) representations of knot groups IV: twisted Alexander polynomials

In this paper we will study properties of twisted Alexander polynomials of knots corresponding to metabelian representations. In particular we answer a question of Wada about the twisted Alexander polynomial associated to the tensor product of two representations, and we settle several conjectures of Hirasawa and Murasugi.

The Thurston norm and twisted Alexander polynomials

Using recent results of Agol, Przytycki-Wise and Wise we show that twisted Alexander polynomials detect the Thurston norm of any irreducible 3-manifold which is not a closed graph manifold.

Twisted Alexander polynomials detect the unknot

The group of a nontrivial knot admits a finite permutation representation such that the corresponding twisted Alexander polynomial is not a unit.

On the Alexander polynomial and the signature invariant of two-bridge knots

Fox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation between the number of the stable coefficients in the Alexander polynomial and the signature invariant. In this paper we prove the Hirasawa-Murasugi conjecture for two-bridge knots.

Twisted Alexander polynomial of links in the projective space

We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.

On a Generalization of Alexander Polynomial for Long Virtual Knots

We construct new invariant polynomial for long virtual knots. It is a generalization of Alexander polynomial. We designate it by $ζ$ meaning an analogy with $ζ$-polynomial for virtual links. A degree of $ζ$-polynomial estimates a virtual crossing number. We describe some application of $ζ$-polynomial for the study of minimal long virtual diagrams with respect number of virtual crossings.

Higher dimensional twisted Alexander polynomials for metabelian representations

We study the asymptotic behavior of the twisted Alexander polynomial for the sequence of SL(n ,C)-representations induced from an irreducible metabelian SL(2, C)-representation of a knot group. We give the limits of the leading coefficients in the asymptotics of the twisted Alexander polynomial and related Reidemeister torsion. The concrete computations for all genus one two-bridge knots are also presented.

Alexander invariants for virtual knots

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 Alexander polynomial, and we show that it is closely related to the generalized Alexander polynomial $G_K(s,t)$ introduced by Sawollek, Kauffman-Radford, and Silver-Williams. We define a natural normalization of the virtual Alexander polynomial and show it satisfies a skein formula. W…

与有限群正则表示相关联的纽结的扭曲Alexander多项式

Twisted Alexander polynomials of knots associated to the regular representations of finite groups

本文研究了与有限群正则表示相关联的纽结扭曲Alexander多项式，给出了通过满射同态与正则表示复合获得的表示下，用Alexander多项式表示的若干公式。

Polynomial cocycles of Alexander quandles and applications

Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all odd dimensions.

Knotoids 与 Linkoids 的 Mock Alexander 多项式

The Mock Alexander Polynomial for Knotoids and Linkoids

本文研究 knotoids 与 linkoids 的 mock Alexander 多项式，证明并推广相关猜想，同时基于广义 knotoids 形式体系构造 linkoids 的规范不变量。

A two component link with Alexander polynomial one is concordant to the Hopf link

Four-dimensional surgery is used to show that a two component link with Alexander polynomial one is topologically concordant to the Hopf link.

The Multivariable Alexander Polynomial for a Closed Braid

A simple multivariable version of the reduced Burau matrix is constructed for any braid. It is shown how the multivariable Alexander polynomial for the closure of the braid can be found directly from this matrix.

结的群扩展的扭曲Alexander多项式

Twisted Alexander polynomials of a knot for group extensions

本文探讨了有限群的群扩展中结的扭曲Alexander多项式，提供了模p的公式，并研究了中心扩展中的多项式应用，用于非纤维结的扭曲Alexander消失群和阶研究。

Alexander-Conway Polynomial State Model and Link Homology

This paper shows how the Formal Knot Theory state model for the Alexander-Conway polynomial is related to Knot Floer Homology. In particular we prove a parity result about the states in this model that clarifies certain relationships of the model with Knot Floer Homology.

On the Links-Gould invariant and the square of the Alexander polynomial

This paper gives a connection between well chosen reductions of the Links-Gould invariants of oriented links and powers of the Alexander-Conway polynomial. We prove these formulas by showing the representations of the braid groups we derive the specialized Links-Gould polynomials from can be seen as exterior powers of copies of Burau representations.

Alexander and writhe polynomials for virtual knots

We give a new interpretation of the Alexander polynomial $Δ_0$ for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, $Δ_0$ determines the writhe polynomial of Cheng and Gao (equivalently, Kauffman's affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.

Alexander polynomials and hyperbolic volume of arborescent links

We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link of any number of components, and by infinitely many such links of at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperbolic link of at least 2 components. For given polynomial, we give also an upper bound on the minimal hyperbolic volume of knots/links, and contrarily, construct knots of arbitrarily large volume, which are arborescent, or have given free genus at least 2.

The Jones and Alexander polynomials for singular links

We extend the state models for Jones and Alexander polynomials of classical links to state models of 2-variable polynomials in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibility of long singular knots.

Twisted Alexander polynomials of genus one two-bridge knots

Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister torsion of the 3-manifold obtained by a Dehn surgery on a genus one two-bridge knot.

Twisted Alexander invariants detect trivial 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 Alexander modules detect trivial links.

4-移动消灭亚历山大多项式

The 4-move kills the Alexander polynomial

本文证明每个纽结通过一系列4-移动和同痕可化为亚历山大多项式平凡的纽结，但4-移动是否为解结操作仍开放。