Anthony Conway, Stefan Friedl, Enrico Toffoli
It is well known that the Blanchfield pairing of a knot can be expressed using Seifert matrices. In this paper, we compute the Blanchfield pairing of a colored link with non-zero Alexander polynomial. More precisely, we show that the Blanchfield pairing of such a link can be written in terms of generalized Seifert matrices which arise from the use of C-complexes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Blanchfield pairings and twisted Blanchfield pairings of torus knots
本文研究了扭结的Blanchfield配对和扭转Blanchfield配对的显式矩阵表示，通过构造链复形和分析主因子来揭示其结构特性。
In this note we use Blanchfield forms to study knots that can be turned into an unknot using a single $\overline{t}_{2k}$ move.
We prove a decomposition formula for twisted Blanchfield pairings of 3-manifolds. As an application we show that the twisted Blanchfield pairing of a 3-manifold obtained from a 3-manifold Y with a representation $ϕ: Z[π_1(Y)] \to R$, infected by a knot J along a curve $η$ with $ϕ(η) \neq 1$, splits orthogonally as the sum of the twisted Blanchfield pairing of Y and the ordinary Blanchfield pairing of the knot J, with the latter tensored up from $Z[t,t^{-1}]$ to R.
We study some comparison between a bilinear cohomology pairing in local coefficients and the Blanchfield pairing of a knot. We show that the former pairing is an $S$-equivalent invariant, and give a criterion to a relation between the two pairings. We also observe that the pairings of some knots are equivalent, and that the pairings of other knots are not equivalent.
We calculate Blanchfield pairings of 3-manifolds. In particular, we give a formula for the Blanchfield pairing of a fibred 3-manifold and we give a new proof that the Blanchfield pairing of a knot can be expressed in terms of a Seifert matrix.
Given a link $L$, the Blanchfield pairing $\operatorname{Bl}(L)$ is a pairing which is defined on the torsion submodule of the Alexander module of $L$. In some particular cases, namely if $L$ is a boundary link or if the Alexander module of $L$ is torsion, $\operatorname{Bl}(L)$ can be computed explicitly; however no formula is known in general. In this article, we compute the Blanchfield pairing of any link, generalizing the aforementioned results. As a corollary, we obtain a new proof that the Blanchfield pairing is hermitian. Finally, we also obtain short proofs of several properties of $\o…
We establish homotopy ribbon concordance obstructions coming from the Blanchfield form and Levine-Tristram signatures. Then, as an application of twisted Alexander polynomials, we show that for every knot K with nontrivial Alexander polynomial, there exists an infinite family of knots that are all concordant to K and have the same Blanchfield form as K, such that no pair of knots in that family is homotopy ribbon concordant.
Blanchfield pairings and Gordian distance
本文利用 Blanchfield 配对为 Gordian 距离建立下界，并证明至少 195 对交叉数不超过 10 的无向非平凡素纽结的 Gordian 距离为 3。
We give a formula for the duality structure of the 3-manifold obtained by doing zero-framed surgery along a knot in the 3-sphere, starting from a diagram of the knot. We then use this to give a combinatorial algorithm for computing the twisted Blanchfield pairing of such 3-manifolds. With the twisting defined by Casson-Gordon style representations, we use our computation of the twisted Blanchfield pairing to show that some subtle satellites of genus two ribbon knots yield non-slice knots. The construction is subtle in the sense that, once based, the infection curve lies in the second derived s…
In a classic paper Zeeman introduced the k-twist spin of a knot K and showed that the exterior of a twist spin fibers over S^1. In particular this result shows that the knot K # -K is doubly slice. In this paper we give a quick proof of Zeeman's result. The k-twist spin of K also gives rise to two metabolizers for K # -K and we determine these two metabolizers precisely.
Twisted Blanchfield pairings and twisted signatures III: Applications
本文提出用 twisted Blanchfield 形式算法计算纽结的 twisted 签名不变量，并在环面纽结上实现，同时利用卫星公式阻碍迭代环面纽结的切片性。
In this survey article we present connections between Picard--Lefschetz invariants of isolated hypersurface singularities and Blanchfield forms for links. We emphasize the unifying role of Hermitian Variation Structures introduced by Némethi.
We define the twisted Blanchfield pairing of a symmetric triad of chain complexes over a group ring Z[G], together with a unitary representation of G over an Ore domain with involution. We prove that the pairing is sesquilinear, and we prove that it is hermitian and nonsingular under certain extra conditions. A twisted Blanchfield pairing is then associated to a 3-manifold together with a decomposition of its boundary into two pieces and a unitary representation of its fundamental group.
This is the first paper in a series of three devoted to studying twisted linking forms of knots and three-manifolds. Its function is to provide the algebraic foundations for the next two papers by describing how to define and calculate signature invariants associated to a linking form $M\times M\to\mathbb{F}(t)/\mathbb{F}[t^{\pm1}]$ for $\mathbb{F}=\mathbb{R},\mathbb{C}$, where $M$ is a torsion $\mathbb{F}[t^{\pm 1}]$-module. Along the way, we classify such linking forms up to isometry and Witt equivalence and study whether they can be represented by matrices.
Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants
本文提出扭曲签名函数及其与Casson-Gordon不变量的关系，通过卫星公式扩展了结同调的研究。
Given a link in $S^3$ we will use invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinking number and various splitting numbers. These lower bounds generalise results recently obtained by Kawauchi. We give an application restricting the knot types which can arise from a sequence of splitting operations on a link. This allows us to answer a question asked by Colin Adams in 1996.
We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots $D^{n-2}\into D^n$. Cobordisms of disk knots that do not fix the boundary sphere knots are easily classified by the cobordism properties of these boundaries, and any two even-dimensional disk knots with isotopic boundary knots are cobordant rel boundary. However, the cobordism rel boundary theory of odd-dimensional disk knots is more subtle. Generalizing results of Levine on cobordism of sphere knots, we define disk knot Seifert matrices and show that two higher-dimensional disk knots with isotopic boundaries are cob…
In this paper we survey some work on representations of $B_n$ given by the induced action on a homology module of some space. One of these, called the Lawrence-Krammer representation, recently came to prominence when it was shown to be faithful for all $n$. We will outline the methods used, applying them to a closely related representation for which the proof is slightly easier. The main tool is the Blanchfield pairing, a sesquilinear pairing between elements of relative homology. We discuss two other applications of the Blanchfield pairing, namely a proof that the Burau representation is not …
On Arf invariants of colored links
本文研究了着色链接的Arf不变量扩展，通过广义Seifert形式构造二次型，并探讨其与链接数的关系。
Using Blanchfield pairings, we show that two Alexander polynomials cannot be realized by a pair of matrices with Gordian distance one if a corresponding quadratic equation does not have an integer solution. We also give an example of how our results help in calculating the Gordian distances, algebraic Gordian distances and polynomial distances.
This paper concerns twisted signature invariants of knots and 3-manifolds. In the fibered case, we reduce the computation of these invariants to the study of the intersection form and monodromy on the twisted homology of the fiber surface. Along the way, we use rings of power series to obtain new interpretations of the twisted Milnor pairing introduced by Kirk and Livingston. This allows us to relate these pairings to twisted Blanchfield pairings. Finally, we study the resulting signature invariants, all of which are twisted generalisations of the Levine-Tristram signature.
Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(K) of this Blanchfield pairing with respect to a framed knot K that generates H_1(M)/Torsion. This article is devoted to the study of the invariant Q. We prove many properties for this invariant including two surgery formulae. Via surgery, the invariant Q is equivalent to an invariant of null-homologous kno…
Recently Kearton showed that any Seifert matrix of a knot is S--equivalent to the Seifert matrix of a prime knot. We show in this note that such a matrix is in fact S--equivalent to the Seifert matrix of a hyperbolic knot. This result follows from reinterpreting this problem in terms of Blanchfield pairings and by applying results of Kawauchi.
Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space
本研究给出纽结Z-切片亏格的三种三维刻画并推广至相关流形情形，推导其有效下界并证明奇素数重分支覆叠环绕配对的等距确定性，最终完成交叉数≤12的所有素纽结的Z-切片亏格计算。
We generalize the works of Lee [arXiv:math/0210213v3] and Gornik [arXiv:math/0402266v2] to construct a basis for generic deformations of the colored sl(N)-homology defined in [arXiv:1002.2662v1]. As applications, we construct non-degenerate pairings and co-pairings which lead to dualities of generic deformations of the colored sl(N)-homology. We also define and study colored sl(N)-Rasmussen invariants. Among other things, we observe that these invariants vanish on amphicheiral knots and discuss some implications of this observation.
Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the configuration space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(M,K) of this Blanchfield pairing with respect to a framed knot K that generates H_1(M;Z)/Torsion. We present the invariant Q(M,K) and some of its properties including a surgery formula. Via surgery, the invariant Q is equivalent to an invariant Q' of null-homologous knots in rational homology …
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。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。