An Explicit Computation of the Blanchfield Pairing for Arbitrary Links

  title={An Explicit Computation of the Blanchfield Pairing for Arbitrary Links},
  author={Anthony Conway},
  journal={Canadian Journal of Mathematics},
  pages={983 - 1007}
  • Anthony Conway
  • Published 1 June 2017
  • Mathematics
  • Canadian Journal of Mathematics
Abstract Given a link $L$ , the Blanchfield pairing $\text{Bl(}L\text{)}$ is a pairing that 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, $\text{Bl(}L\text{)}$ 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… 

Twisted Blanchfield pairings, twisted signatures and Casson-Gordon invariants

This paper decomposes into two main parts. In the algebraic part, we prove an isometry classification of linking forms over $\mathbb{R}[t^{\pm 1}]$ and $\mathbb{C}[t^{\pm 1}]$. Using this result, we

The C-complex clasp number of links

It is proved that if $L$ is a 2-component link with nonzero linking number, then the linking number determines the minimal number of clasps amongst all C-complexes.

Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants

. This paper studies twisted signature invariants and twisted linking forms, with a view towards obstructions to knot concordance. Given a knot K and a representation ρ of the knot group, we define a

Untwisting number and Blanchfield pairings

In this note we use Blanchfield forms to study knots that can be turned into an unknot using a single $\overline{t}_{2k}$ move.

The Z -genus of boundary links

It is shown that a variant of the shake genus of a knot, the Z -shake genus, equals the Z-genus of the knot.

An Algorithm to Calculate Generalized Seifert Matrices

An algorithm for computing generalized Seifert matrices for colored links given as closures of colored braids is developed and implemented by the second author as a computer program called Clasper.

Abelian invariants of doubly slice links

We provide obstructions to a link in S3 arising as the cross section of any number of unlinked spheres in S4. Our obstructions arise from the multivariable signature, the Blanchfield form and

The $$\mathbb Z$$-genus of boundary links

<jats:p>The <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb Z$$</jats:tex-math><mml:math xmlns:mml=""> <mml:mi>Z</mml:mi>



Twisted Blanchfield pairings and decompositions of 3-manifolds

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

Invariants of boundary link cobordism II

We use the Blanchfield-Duval form to define complete invariants for the cobordism group C2q−1(Fμ) of (2q − 1)-dimensional μ-component boundary links (for q ≥ 2). The author solved the same problem in

Blanchfield duality and simple knots

The method of presentation for n-knots is used to classify simple (2q l)-knots, q > 3, in terms of the Blanchfield duality pairing. As a corollary, we characterize the homology modules and pairings

Classification of simple knots by Blanchfield duality

0. Introduction. The purpose of this paper is to announce some results on simple knots, i.e. knots of S~ in S + 1 whose complements have the homotopy (q — l)-type of S. We state in §4 two theorems

Blanchfield and Seifert algebra in high dimensional knot theory

Novikov [12] initiated the study of the algebraic properties of quadratic forms over polynomial extensions by a far-reaching analogue of the Pontrjagin-Thom transversality construction of a Seifert

The universal abelian cover of a link

Introduction Given a Seifert surface for a classical knot, there is associated a linking form from which the first homology of the infinite cyclic cover may be obtained. This article considers

Rational Blanchfield forms, S-equivalence, and null LP-surgeries

Null Lagrangian-preserving surgeries are a generalization of the Garoufalidis and Rozansky null-moves, that these authors introduced to study the Kricker lift of the Kontsevich integral, in the

Knot modules. I

For a differentiable knot, i.e. an imbedding SI C S,+2, one can associate a sequence of modules (Aq) over the ring Z [t, I -l], which are the source of many classical knot invariants. If X is the

The Blanchfield pairing of colored links

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