Oleg Viro
The multivariable Conway function is generalized to oriented framed trivalent graphs equipped with additional structure (coloring). This is done via refinements of Reshetikhin-Turaev functors based on irreducible representations of quantized gl(1|1) and sl(2). The corresponding face state sum models for the generalized Conway function are presented.
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We give an explicit formula of the Alexander polynomial of the link obtained by adding an arbitrary number of full twists to positively oriented parallel n-strands in terms of the Alexander polynomials of the links obtained by adding 0,1,...,n-1 full twists. From this, we see that the Alexander polynomials stabilize after adding sufficiently many full twists. The main tool used in the computation is expressing the Alexander polynomial using the vector space representation of $U_{q}(gl(1|1))$.
In this note, we consider the problem of constructing knot invariants from Yang-Baxter operators associated to (unitary associative) algebra structures. We first compute the enhancements of these operators. Then, we conclude that Turaev's procedure to construct knot invariants, as modified by Murakami, invariably produces from those enhanced Yang-Baxter operators the Alexander polynomial.
Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov--Rozansky.
We develop a diagrammatic calculus for representations of unrolled quantum $\mathfrak{sl}_2$ at a fourth root of unity. This allows us to prove Seifert-Torres type formulas for certain splice links using quantum algebraic methods, rather than topological methods. Other applications of this diagrammatic calculus given here are a skein relation for $n$-cabled double crossings and a simple proof that the quantum invariant associated with these representations determines the multivariable Alexander polynomial.
We use the 2-loop term of the Kontsevich integral to show that there are (many) knots with trivial Alexander polynomial which don't have a Seifert surface whose genus equals the rank of the Seifert form. This is one of the first applications of the Kontsevich integral to intrinsically 3-dimensional questions in topology. Our examples contradict a lemma of Mike Freedman, and we explain what went wrong in his argument and why the mistake is irrelevant for topological knot concordance. References updated.
Oleg Viro studied in arXiv:math/0204290 two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to $U_q sl(2)$ at fourth roots of unity, or by considering the super Hopf algebra $U_q gl(1|1)$. In this paper, we show these Hopf algebras share properties with the $-1$ specialization of $U_q gl(n|1)$ leading to the proof of a conjecture of David De Wit, Atsushi Ishii and Jon Links on the Links-Gould invariants.
Using the Fourier expansion of Markov traces for Ariki-Koike algebras over $\mathbb{Q}(q,u_{1},...,u_{e})$, we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of a Markov parameter, the Fourier coefficients of Markov traces take quite simple form. As a consequence, we show that the Alexander polynomial of a mixed link is essentially equal to the Alexander polynomial of the link obtained by resolving the twisted parts.
Defined by Joyce and Matveev, the fundamental quandle is a complete invariant of oriented classical knots. We consider invariants of knots defined from quotients of the fundamental quandle. In particular, we introduce the fundamental Latin Alexander quandle of a knot and consider its Gröbner basis-valued invariants, which generalize the Alexander polynomial. We show via example that the invariant is not determined by the generalized Alexander polynomial for virtual knots.
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 $μ_{i_1... i_p}$ vanish for $p\le n$. We express this coefficient as a polynomial in Milnor numbers of L. Depending on whether the parity of n is odd or even, the terms in this polynomial correspond either to spanning trees in certain graphs or to decompositions of certain 3-graphs into pairs of spa…
We connect Dedekind sums and Alexander polynomials of torus knots.
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.
We give a formula for Alexander polynomials of doubly primitive knots.
We calculate the twisted Alexander polynomials of all tunnel number one Montesinos knots associated to their $SL_2(\mathbb{C})$-representations and obtain their leading coefficients and degrees. As a corollary, we get some interesting examples, that is, nonfibered knots with monic Alexander polynomials which have non-monic twisted Alexander polynomials.
We prove duality theorems for twisted Reidemeister torsions and twisted Alexander polynomials generalizing the results of Turaev. As a corollary we determine the parity of the degrees of twisted Alexander polynomials of 3-manifolds in many cases.
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.
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。
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.
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.
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…
We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 $p^n$.
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.
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.
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.
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.
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。
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, under certain assumptions, the twisted Alexander modules of a complex hypersurface are torsion modules, and that their orders, the twisted Alexander polynomials, divide the product of certain `local polynomials' defined in terms of the topology near singularities.
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.
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.
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.
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 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.
This paper has been withdrawn because the result turns out to be trivial.
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.
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。
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。
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).
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…
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.
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.
The group of a nontrivial knot admits a finite permutation representation such that the corresponding twisted Alexander polynomial is not a unit.
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.
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 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.
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.
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…
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。
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 intersection Alexander polynomials satisfy certain duality and normalization conditions analogous to those of ordinary Alexander polynomials, and we explore the relationships between the intersection Alexander polynomials and certain generalizations of the classical Alexander poly…
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.
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