David Cimasoni
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.
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We calculate the $L^2$-Alexander torsion for Seifert fiber spaces and graph manifolds in terms of the Thurston norm.
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.
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.
We show that bordered Floer homology provides a categorification of a TQFT described by Donaldson. This, in turn, leads to a proof that both the Alexander module of a knot and the Seifert form are completely determined by Heegaard Floer theory.
The coefficients of twisted Alexander polynomials of a knot induce regular functions of the $SL_2(\mathbb{C})$-character variety. We prove that the function of the highest degree has a finite value at an ideal point which gives a minimal genus Seifert surface by Culler-Shalen theory. It implies a partial affirmative answer to a conjecture by Dunfield, Friedl and Jackson.
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.
Alexander polynomial, Dijkgraaf-Witten invariant, and Seifert fibred surgery
本文利用 Dijkgraaf-Witten 不变量研究沿纽结的 k/ℓ 手术产生的小 Seifert 3-流形，证明其重数与手术系数受 Alexander 多项式约束。
We explain an algorithm for finding a boundary link Seifert matrix for a given Alexander polynomial. The algorithm depends on several choices and therefore makes it possible to find non-equivalent Seifert matrices for a given Alexander polynomial.
On elementary invariants of genus one knots and Seifert surfaces
本文提出了一种用于研究一阶结和塞弗特面的亚历山大形式不变量，通过分析三维同调柄体的性质来证明其不变性。
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. We illustrate examples of knots that have trivial Alexander polynomials but do not have twisted Alexander invariants of fibred knots.
We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance group and not concordant to any knot with coprime Alexander polynomial. Key ingredients for the proof are Cheeger-Gromov-von Neumann rho-invariants for amenable groups developed by Cha and Orr and polynomial splittings of metabelian rho-invariants.
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.
In this paper we construct a multivariable link invariant arising from the quantum group associated to the special linear Lie superalgebra sl(2|1). The usual quantum group invariant of links associated to (generic) representations of sl(2|1) is trivial. However, we modify this construction and define a nontrivial link invariant. This new invariant can be thought of as a multivariable version of the Links-Gould invariant. We also show that after a variable reduction our invariant specializes to the Conway potential function, which is a version of the multivariable Alexander polynomial.
We develop an algorithm for computing generalized Seifert matrices for colored links given as closures of colored braids. The algorithm has been implemented by the second author as a computer program called Clasper. Clasper also outputs the Conway potential function, the multivariable Alexander polynomial and the Cimasoni-Florens signatures of a link, and displays a visualization of the C-complex used for producing the generalized Seifert matrices.
We explore certain restrictions on knots in the three-sphere which admit non-trivial Seifert fibered surgeries. These restrictions stem from the Heegaard Floer homology for Seifert fibered spaces, and hence they have consequences for both the Alexander polynomial of such knots, and also their knot Floer homology. In particular, we show that certain polynomials are never the Alexander polynomials of knots which admit homology three-sphere Seifert fibered surgeries. The knot Floer homology restrictions, on the other hand, apply also in cases where the Alexander polynomial gives no information, s…
We study various specializations of the colored HOMFLY-PT polynomial. These specializations are used to show that the multivariable link invariants arising from a complex family of sl(m|n) super-modules previously defined by the authors contains both the multivariable Alexander polynomial and Kashaev's invariants. We conjecture these multivariable link invariants also specialize to the generalized multivariable Alexander invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki.
We introduce generalized arrow diagrams and generalized Reidemeister moves for diagrams of links in Seifert fibered spaces. We give a presentation of the fundamental group of the link complement. As a corollary we are able to compute the first homology group of the complement and the twisted Alexander polynomials of the link.
We show two results about the Conway potential function which is known as the normalized multivariable Alexander polynomial. We first show that the Conway potential function introduced by Kauffman in "Formal Knot Theory" is indeed a link invariant. Next we show that Kauffman's potential function equals Hartley's potential function. We will prove it by using Murakami's axioms for 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.
For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the image of the commutator subgroup of the knot group.
Let K be a knot in S^3 of genus g and let n>0. We show that if rk HFK(K,g) < 2^{n+1} (where HFK denotes knot Floer homology), in particular if K is an alternating knot such that the leading coefficient a_g of its Alexander polynomial satisfies |a_g| <2^{n+1}, then K has at most n pairwise disjoint non-isotopic genus g Seifert surfaces. For n=1 this implies that K has a unique minimal genus Seifert surface up to isotopy.
In this paper we find a family of knots with trivial Alexander polynomial, and construct two non-isotopic Seifert surfaces for each member in our family. In order to distinguish the surfaces we study the sutured Floer homology invariants of the sutured manifolds obtained by cutting the knot complements along the Seifert surfaces. Our examples provide the first use of sutured Floer homology, and not merely its Euler characteristic(a classical torsion), to distinguish Seifert surfaces. Our technique uses a version of Floer homology, called Longitude Floer Homology in a way that enables us to byp…
A virtual knot that has a homologically trivial representative $\mathscr{K}$ in a thickened surface $Σ\times [0,1]$ is said to be an almost classical (AC) knot. $\mathscr{K}$ then bounds a Seifert surface $F\subset Σ\times [0,1]$. Seifert surfaces of AC knots are useful for computing concordance invariants and slice obstructions. However, Seifert surfaces in $Σ\times [0,1]$ are difficult to construct. Here we introduce virtual Seifert surfaces of AC knots. These are planar figures representing $F \subset Σ\times [0,1]$. An algorithm for constructing a virtual Seifert surface from a Gauss diagr…
The flat plumbing basket presentation of a link is introduced by Furihata, Hirasawa and Kobayashi. In this paper, we show that the pass-equivalence and an equivalence introduced by using the flat plumbing basket presentation are the same relation. Furthermore, we obtain an evaluation of the minimal number of bands used for the flat pluming basket presentation for a knot coming from the degree of the Alexander polynomial and the three genus of the knot.
A diagrammatic computation of abelian link invariants
本文提出通过单个对称矩阵计算着色链接的多变量签名和alexander多项式，并扩展了kauffman的行列式模型。
The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces of this endomorphism. We show that the Euler characteristics of these generalized eigenspaces are the coefficients of the Alexander polynomial of the knot. Among other applications, we deduce that instanton homology detects fibered knots.
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。
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.
In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in S^3. This yields an integral valued function on the m-dimensional torus, where m is the number of colors of the link. The case m=1 corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this m-variable generalization: it vanishes for achiral colored links, it is `piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a 4-dimensional interpretation and th…
Knot Floer homology is an invariant for knots in the three-sphere for which the Euler characteristic is the Alexander-Conway polynomial of the knot. The aim of this paper is to study this homology for a class of satellite knots, so as to see how a certain relation between the Alexander-Conway polynomials of the satellite, companion and pattern is generalized on the level of the knot Floer homology. We also use our observations to study a classical geometric invariant, the Seifert genus, of our satellite knots.
We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured F…
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.
A lower bound for the genus of a knot using the Links-Gould invariant
通过研究Links-Gould多项式次数，给出了任意纽结Seifert亏格的下界，改进了Alexander多项式的Seifert不等式，并检测到Kinoshita-Terasaka和Conway纽结的亏格至少为2。
Algebraic properties of twisted Alexander polynomial and Reidemeister torsion of torus knots
本文证明了扭Alexander多项式中与不可约SL_n(C)表示相关的每个系数都是SL_n(C)特征标簇上的A-值局部常函数，其中A是所有复数上的代数整数环。此外，作为Kitano和Nozaki最近结果的推广，我们显示SL_n(C)的Reidemeister扭转对于许多Seifert纤维空间是代数整数。同时，我们讨论了扭结的Reidemeister扭转的幂和，这些扭结的低维不可约表示与TQFT存在神秘关系。
We show that for a special alternating link diagram, the following three polynomials are essentially the same: a) the part of the HOMFLY polynomial that corresponds to the leading term in the Alexander polynomial; b) the $h$-vector for a triangulation of the root polytope of the Seifert graph and c) the enumerator of parking functions for the planar dual of the Seifert graph. These observations yield formulas for the maximal $z$-degree part of the HOMFLY polynomial of an arbitrary homogeneous link as well. Our result is part of a program aimed at reading HOMFLY coefficients out of Heegaard Flo…
Spectral Factorization and Hypergeometric Representations of the Alexander Polynomials of $Th(4,2n+1)$
研究4股土耳其头结$Th(4,2n+1)$的亚历山大多项式，通过Burau表示导出递推关系和生成函数，利用Chebyshev多项式分解得到超几何级数表示，并分析渐近性质。
Mock Seifert matrices and unoriented algebraic concordance
本文通过Mock Seifert矩阵定义无向代数协边群，证明其阿贝尔且无限生成，并用于研究虚拟纽结协边等不变量。
Given a transverse knot $K$ in a three dimensional contact manifold $(Y,α)$, in [13] Colin, Ghiggini, Honda and Hutchings define a hat version of embedded contact homology for $K$, that we call $\widehat{ECK}(K,Y,α)$, and conjecture that it is isomorphic to the knot Floer homology $\widehat{HFK}(K,Y)$. We define here a full version $ECK(K,Y,α)$ and generalise the definitions to the case of links. We prove then that, if $Y = S^3$, $ECK$ and $\widehat{ECK}$ categorify the (multivariable) Alexander polynomial of knots and links, obtaining expressions analogue to that for knot and link Floer homol…
We introduce a new invariant of tangles along with an algebraic framework in which to understand it. We claim that the invariant contains the classical Alexander polynomial of knots and its multivariable extension to links. We argue that of the computationally efficient members of the family of Alexander invariants, it is the most meaningful. These are lecture notes for talks given by the first author, written and completed by the second. The talks, with handouts and videos, are available at http://www.math.toronto.edu/drorbn/Talks/Regina-1206/. See also further comments at http://www.math.tor…
We study generalizations of a classical link invariant -- the multivariable Alexander polynomial -- to tangles. The starting point is Archibald's tMVA invariant for virtual tangles which lives in the setting of circuit algebras, and whose target space has dimension that is exponential in the number of strands. Using the Hodge star map and restricting to tangles without closed components, we define a reduction of the tMVA to an invariant "rMVA" which is valued in matrices with Laurent polynomial entries, and so has a much more compact target space. We show the rMVA has the structure of a metamo…
S-Equivalence of Band-Twisted Genus One Knots
本文构造了一列亏格一结，它们彼此 $S$-等价，但可通过 Jones 多项式区分，回应了 Kirby 问题列表中的问题 1.6 (K3)。
New invariants for virtual knots via spanning surfaces
该研究定义了加厚曲面上纽结的三类张成曲面，提出新的Seifert矩阵、Alexander型多项式等不变量，构造了加厚曲面上纽结的Heegaard Floer同调并给出虚拟纽结的Floer同调定义，还用δ-不变量阻碍纽结的稳定化关系。
Multivariable knot polynomials from braided Hopf algebras with automorphisms
本文通过带自同构的辫子 Hopf 代数上的 Yetter--Drinfel'd 模求解 Yang--Baxter 方程，构造出包含 ADO、着色 Jones 及新 2 变量多项式在内的多元纽结不变量，新多项式可检测手性与突变并给出紧致亏格界。
Extended signatures and link concordance
本文研究了扩展Levine-Tristram签名作为连通性不变量，并提出了一个无法被传统方法检测的3链环族。
It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomial-ribbon genus conjectures for a quasipositive knot. Then we show that the Alexander polynomial completely detects the minimal genus and fiber property of canonical Seifert surfaces associated to almost positive (and almost alternating) link diagrams.
Algebraic concordance groups for links
研究利用广义Seifert矩阵族定义链环的代数协边群，借此恢复相关不变性，并用于研究群的同构类型。
$U_q(\mathfrak{gl}(m|n))$ bounds on the minimal genus of virtual links
本文基于量子超群$U_q(\mathfrak{gl}(m|n))$提出虚拟链接最小亏格的下界，改进了传统方法，并应用于结的亏格性质研究。
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