# An Explicit Computation of the Blanchfield Pairing for Arbitrary Links

@article{Conway2018AnEC, title={An Explicit Computation of the Blanchfield Pairing for Arbitrary Links}, author={Anthony Conway}, journal={Canadian Journal of Mathematics}, year={2018}, volume={70}, pages={983 - 1007} }

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…

## 9 Citations

### Twisted Blanchfield pairings, twisted signatures and Casson-Gordon invariants

- Mathematics
- 2018

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

- Computer ScienceRocky Mountain Journal of Mathematics
- 2020

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 I: Algebraic background

- MathematicsLinear Algebra and its Applications
- 2022

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

- Mathematics
- 2018

. 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

- Mathematics
- 2017

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

- Biology, Mathematics
- 2022

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

- Computer ScienceJournal of Knot Theory and Its Ramifications
- 2022

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

- MathematicsL’Enseignement Mathématique
- 2022

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

- MathematicsRevista Matemática Complutense
- 2022

<jats:p>The <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb Z$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>Z</mml:mi>…

## References

SHOWING 1-10 OF 54 REFERENCES

### Twisted Blanchfield pairings and decompositions of 3-manifolds

- Mathematics
- 2016

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

- Mathematics
- 2006

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

- Mathematics
- 1975

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

- Mathematics
- 1973

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

- Mathematics
- 2002

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

- Mathematics
- 1982

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

- Mathematics
- 2012

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…

### HOMOLOGY BOUNDARY LINKS AND BLANCHFIELD FORMS: CONCORDANCE CLASSIFICATION AND NEW TANGLE-THEORETIC CONSTRUCTIONS

- Mathematics
- 1994

### Knot modules. I

- Mathematics
- 1977

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

- Mathematics
- 2016

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…