Eva Horvat, Boštjan Gabrovšek
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.
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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.
Ozsváth-Szabó proved the property that any coefficient of Alexander polynomial of lens space knot is either $\pm1$ or $0$ and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsváth-Szabó's property as the existence of simple curves included in a region in ${\Bbb R}^2$. The existence of curves, that has no end-points connected, is just 1-component in a region, can search distribution of non-zero coefficients of the Alexander polynomial of the lens space knot. This cu…
We give an alternative proof of a recent theorem of Tange using the technology of changemaker lattices. Specifically, for $K\subset S^3$ a non-trivial knot with a lens space surgery, we give constraints on the Alexander polynomial of $K$ and the lens space surgery when the coefficient of $T^{g(K)-2}$ in $Δ_K(T)$ is non-zero.
$\mathfrak{gl}(1 \vert 1)$-Alexander polynomial for $3$-manifolds, Reidemeister torsion and lens spaces
本文重新表述了先前定义的不变量$Δ(M, ω)$，揭示了其与Reidemeister挠率的关系，并在$3$-流形为透镜空间时计算了该不变量，进而探讨了利用它对透镜空间的分类。
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.
We characterize the first Alexander Z[Z]-modules of ribbon surface-links in the 4-sphere fixing the number of components and the total genus, and then the first Alexander Z[Z]-modules of surface-links in the 4-sphere fixing the number of components. Using the result of ribbon torus-links, we also characterize the first Alexander Z[Z]-modules of virtual links fixing the number of components. For a general surface-link, an estimate of the total genus is given in terms of the first Alexander Z[Z]-module. We show a graded structure on the first Alexander Z[Z]-modules of all surface-links and then …
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。
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 exceptional cases, by showing that the link concordance class is not determined by the Alexander invariants in any other case.
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 these links can be easily derived.
Four-dimensional surgery is used to show that a two component link with Alexander polynomial one is topologically concordant to the Hopf link.
We give infinitely many $2$-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any $2$-component link with trivial Alexander polynomial. Our examples are pairwise non-concordant.
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.
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 Alexander quandle $Q$, also show that if $Δ_{L}(t)\not=0, 1$, then $L$ admits a non-trivial coloring by the Alexander quandle $Λ/(Δ_{L}(t))$.
A. Ishii and K. Oshiro introduced the notion of an $f$-twisted Alexander matrix. This notion is a quandle version of the twisted Alexander matrix which was introduced by M. Wada. They showed that the twisted Alexander matrix of a pair of a link group and a group representation can be recovered from the $f$-twisted Alexander matrix of a pair of a link quandle and a quandle homomorphism. In this paper, we study a relationship between the $f$-twisted Alexander matrix and the quandle cocycle invariant. As an application, we show that an $f$-twisted Alexander invariant of knot quandles is a really …
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 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.
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.
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。
The multivariate Alexander module of a link L has several subsets that admit quandle operations defined using the module operations. One of them, the fundamental multivariate Alexander quandle, determines the link module sequence of L.
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.
Joyce observed that the Alexander invariant and the medial quandle of a classical knot are equivalent to each other, as invariants. In the present paper, we discuss the rather complicated extension of Joyce's observation to several different medial quandles and reduced (one-variable) Alexander modules associated with classical links. The theme is that for links, medial quandles provide stronger invariants than reduced Alexander modules.
Walter Neumann showed that the topology of a ``regular'' algebraic curve V in C^2 is determined up to proper isotopy by some link in S^3 called the link at infinity of V. In this note, we compute the Alexander module over C[t^{\pm 1}] of any such link at infinity.
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.
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.
There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links, and observe their behaviors. For spatial graphs, we calculate the invariants of Suzuki's theta-curves and show that the invariants are nontrivial for Suzuki's theta-curves whose Alexander ideals are trivial. For handlebody-knots, we give a remark on…
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.
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.
We consider Conway polynomials of two-bridge links as Euler continuant polynomials. As a consequence, we obtain new and elementary proofs of classical Murasugi's 1958 alternating theorem and Hartley's 1979 trapezoidal theorem. We give a modulo 2 congruence for links, which implies the classical Murasugi's 1971 congruence for knots. We also give sharp bounds for the coefficients of Euler continuants and deduce bounds for the Alexander polynomials of two-bridge links. These bounds improve and generalize those of Nakanishi Suketa'96. We easily obtain some bounds for the roots of the Alexander pol…
We prove a Torres-like formula for the $L^2$-Alexander torsions of links, as well as formulas for connected sums and cablings of links. Along the way we compute explicitly the $L^2$-Alexander torsions of torus links inside the three-sphere, the solid torus and the thickened torus.
A well-known identity (Alex+) - (Alex-)=(t^{1/2}-t^{-1/2}) (Alex0) holds for three 1-links L+, L-, and L0 which satisfy a famous local-move-relation. We prove a new local-move-identity for the Z[t,t^{-1}]-Alexander polynomials of 2-links, which is a 2-dimensional analogue of the 1-dimensional one. In the 1-dimensional link case there is a well-known relation between the Alexander-Conway polynomial and the linking number. As its 2-dimensional analogue, we find a relation between the Z[t,t^{-1}]-Alexander polynomials of 2-links and the alinking number of 2-links. We show high dimensional analogu…
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.
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。
In this paper we present a sequence of link invariants, defined from twisted Alexander polynomials, and discuss their effectiveness in distinguish knots. In particular, we recast and extend by geometric means a recent result of Silver and Williams on the nontriviality of twisted Alexander polynomials for nontrivial knots. Furthermore building on results in [FV06b] we prove that these invariants decide if a genus one knot is fibered, and we also show that these invariants distinguish all mutants with up to 12 crossings.
Gröbner Bases for Alexander Fitting Ideals of Links
针对多变量Alexander理论生成的无优选生成元的Fitting理想，固定系数域与单项序后，通过多项式收缩的简约Gröbner基表示这些理想得到纽结的Alexander-Gröbner不变量，还推导了行列式-除子互反性并计算了多种纽结的相关不变量。
We determine lens surgeries (i.e.\ Dehn surgery yielding a lens space) along the $n$-twisted Whitehead link. To do so, we first give necessary conditions to yield a lens space from the Alexander polynomial of the link as: (1) $n=1$ (i.e. the Whitehead link), and (2) one of surgery coefficients is 1, 2 or 3. Our interests are not only lens surgery itself but also how to apply the Alexander polynomial for this kind of problems.
Multivariate Alexander quandles, VI. Metabelian groups and 2-component links
本文研究了多变量亚历山大淬火与可解群的关系，证明了其与链接群商群的同构性，并揭示了二链环的 medial 淬火与简化亚历山大不变量的联系。
It is well-known that the second coefficient of the Alexander polynomial of any lens space knot in $S^3$ is $-1$. We show that the non-zero third coefficient condition of the Alexander polynomial of a lens space knot $K$ in $S^3$ confines the surgery to the one realized by the $(2,2g+1)$-torus knot, where $g$ is the genus of $K$. In particular, such a lens surgery polynomial coincides with $Δ_{T(2,2g+1)}(t)$.
In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic knots to hyperbolic links, and confirm it for an infinite family of hyperbolic 2-bridge links. Moreover we consider a similar problem for parabolic representations of 2-bridge link groups.
Multivariate Alexander quandles, V. Constructing the medial quandle of a link
本文提出通过减少的亚历山大模的周边结构来构建链接的媒介淬火方法。
Let $Δ_{L,ρ_n}(t)$ be the twisted Alexander polynomial with respect to the representation given by the composition of the lift of the holonomy representation of a certain hyperbolic link $L$ and the $n$-dimensional irreducible complex representation of $\text{SL}(2,\mathbb C)$. We consider a sequence of $Δ_{L,ρ_n}(t)$ and extract the volume of the complement of $L$ from the asymptotic behaviour of the sequence obtained by evaluating $t=1$ or $t=-1$.
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.
Peripheral elements in reduced Alexander modules
研究通过增强缩减亚历山大模中的外围元素，提供了一个能确定链接中所有链接数的更强不变量。
We give a geometric proof of the following result of Juhasz. \emph{Let $a_g$ be the leading coefficient of the Alexander polynomial of an alternating knot $K$. If $|a_g|<4$ then $K$ has a unique minimal genus Seifert surface.} In doing so, we are able to generalise the result, replacing `minimal genus' with `incompressible' and `alternating' with `homogeneous'. We also examine the implications of our proof for alternating links in general.
We prove a geometric refinement of Alexander duality for certain 2-complexes, the so-called gropes, embedded into 4-space. This refinement can be roughly formulated as saying that 4-dimensional Alexander duality preserves the disjoint Dwyer filtration. In addition, we give new proofs and extended versions of two lemmas of Freedman and Lin which are of central importance in the A-B-slice problem, the main open problem in the classification theory of topological 4-manifolds. Our methods are group theoretical, rather than using Massey products and Milnor μ-invariants as in the original proofs.
The second coefficient of the Alexander polynomial as a satellite obstruction
本研究引入含正辫子链环等的链环集合，证明P中链环的Alexander多项式首项与第二系数符号相反，以此作为障碍证明部分卫星链环不在P中。
We use the Bar-Natan Zh-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Zh-map is functorial under concordance, and also that Satoh's Tube map (from welded links to ribbon knotted tori in $S^4$) is functorial under concordance. In addition, we extend classical results of Chen, Milnor, and Hillman on the lower central series of link groups to links in thickened surfaces. Our main result is that the generalized Alexander polynomial vanishes on any knot in a thickened surface which is …
The Alexander and Markov theorems for strongly involutive links
本文针对三维球面中的强对合链环，构建回文辫的等变闭包映射并证明其满射性，同时建立对应等变Markov变换，推广了纽结理论中经典的Alexander与Markov定理。
Alexander-Conway and Bracket Polynomials of Pretzel Links $\boldsymbol{P(1,1,n)}$
本文研究三股 pretzel 链环 P(1,1,n) 的 Alexander-Conway 与 Kauffman bracket 多项式，重新考察这两个经典多项式不变量，以深化对其拓扑和代数性质的理解。
We give a new construction of the one-variable Alexander polynomial of an oriented knot or link, and show that it generalizes to a vector valued invariant of oriented tangles.
Let $L$ be a oriented link such that $Σ_n(L)$, the $n$-fold cyclic cover of $S^3$ branched over $L$, is an L-space for some $n \geq 2$. We show that if either $L$ is a strongly quasipositive link other than one with Alexander polynomial a multiple of $(t-1)^{2g(L) + (|L|-1)}$, or $L$ is a quasipositive link other than one with Alexander polynomial divisible by $(t-1)^{2g_4(L) + (|L|-1)}$, then there is an integer $n(L)$, determined by the Alexander polynomial of $L$ in the first case and the Alexander polynomial of $L$ and the smooth $4$-genus of $L$, $g_4(L)$, in the second, such that $n \leq…
We relate some terms on the boundary of the Newton polygon of the Alexander polynomial $Δ(x,y)$ of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize $Δ(x,y)$ so that no $x^{-1}$ or $y^{-1}$ terms appear, but $x^{-1}Δ(x,y)$ and $y^{-1}Δ(x,y)$ have negative exponents, and so that terms of even total degree are positive and terms with odd total degree are negative. If the rational link has a reduced alternating diagram with no self crossings, then $Δ(-1, 0) = 1$. If the standard form of the rational link has $m$…
In this paper we study some aspects of knots and links in lens spaces. Namely, if we consider lens spaces as quotient of the unit ball $B^{3}$ with suitable identification of boundary points, then we can project the links on the equatorial disk of $B^{3}$, obtaining a regular diagram for them. In this contest, we obtain a complete finite set of Reidemeister type moves establishing equivalence, up to ambient isotopy, a Wirtinger type presentation for the fundamental group of the complement of the link and a diagrammatic method giving the first homology group. We also compute Alexander polynomia…
Let L be an oriented (d+1)-component link in the 3-sphere, and let L(q) be the d-component link in a homology 3-sphere that results from performing 1/q-surgery on the last component. Results about the Alexander polynomial and twisted Alexander polynomials of L(q) corresponding to finite-image representations are obtained. The behavior of the invariants as q increases without bound is described.
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 elementary properties of clasper surgery, and a different invariant, the tree-part of the LMO invariant. Our method also constructs links with trivial higher Alexander modules and nontrivial Milnor invariants.
In a recent paper, Allen and Swenberg investigated which link polynomials are capable of detecting causality in (2+1)-dimensional globally hyperbolic spacetimes. They ultimately suggested it is likely that the Jones Polynomial accomplishes this, while the Alexander-Conway polynomial, independently, is insufficient. As the Alexander-Conway polynomial on its own is not likely to detect causality an additional piece of information must be supplemented to the link polynomial. This paper aims to examine the ability of Alexander quandles, to distinguish the connected sum of two Hopf links and Allen-…
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.
We extend several classical invariants of links in the 3-sphere to links in so-called quasi-cylinders. These invariants include the linking number, the Seifert form, the Alexander module, the Alexander-Conway polynomial and the Murasugi-Tristram-Levine signatures.
As an extension of Reshetikhin and Turaev's invariant, Costantino, Geer and Patureau-Mirand constructed $3$-manifold invariants in the setting of relative $G$-modular categories, which include both semisimple and non-semisimple ribbon tensor categories as examples. In this paper, we follow their method to construct a $3$-manifold invariant from Viro's $\mathfrak{gl}(1\vert 1)$-Alexander polynomial. We take lens spaces $L(7, 1)$ and $L(7, 2)$ as examples to show that this invariant can distinguish homotopy equivalent manifolds.
Let C_T be the subgroup of the smooth knot concordance group generated by topologically slice knots and let C_D be the subgroup generated by knots with trivial Alexander polynomial. We prove the quotient C_T/C_D is infinitely generated, and uncover similar structure in the 3-dimensional rational spin bordism group. Our methods also lead to the construction of links that are topologically, but not smoothly, concordant to boundary links.
On Correspondences between the Alexander Polynomials of Special Alternating Links and MOY Graphs
本文建立了特殊交错链环的经典Alexander多项式与MOY图多项式之间的对应关系，证明两者次数及首末系数一致，且MOY系数始终支配经典系数。
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