Abstract

Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral three-fold cover of S^3 branched along a knot alpha subset S^3. Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot alpha can be derived from dihedral covers of alpha . The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications.

Highlights

  • The study of linking numbers between knots in S3 dates back at least as far as Gauss [13]

  • Let the three-manifold M be presented as a three-fold irregular dihedral branched cover of S3, branched along a knot

  • In Theorem 1.2, we give a formula for the linking number in M between any two connected components of the pre-images of γ and δ, in the case where the pre-images of γ and δ have three connected components each

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Summary

Introduction

The study of linking numbers between knots in S3 dates back at least as far as Gauss [13]. Any 3-coloring of α determines an irregular dihedral three-fold covering map f : M → S3 with branching set α, as reviewed in Sect. Given such a three-fold cover f , the preimage of the branching set, f −1(α), has two connected components whose linking number, in M, is either a rational number or undefined. Our results extend the classical linking number computation to include linking numbers of curves other than the branch curves, namely, closed connected components of f −1(γ ) and f −1(δ), where γ , δ ⊂ S3 are curves in the complement of the branching set. Since every closed, connected, oriented three-manifold admits a presentation as a three-fold dihedral cover of S3 branched along a knot, our methods compute all well-defined linking numbers in all three-manifolds; this is proven at the end of Sect. Since every closed, connected, oriented three-manifold admits a presentation as a three-fold dihedral cover of S3 branched along a knot, our methods compute all well-defined linking numbers in all three-manifolds; this is proven at the end of Sect. 1.1

Algorithm Overview and the Main Theorem
Overview of the Article
Irregular Dihedral Covers
The Cell Structure on S3
The Cell Structure on M
Constructing 2-Chains Bounding Pseudo-Branch Curves
A Note on Pseudo-Branch Curves Which Lift to Fewer Than Three Loops
Linking Numbers Between Branch Curves
Linking Numbers Between Branch and Pseudo-Branch Curves
Examples
Characteristic Knots

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