Linking Numbers in Three-Manifolds

TL;DR

This paper presents an explicit algorithm to compute linking numbers in any closed, oriented three-manifold using dihedral covers, aiding in knot theory and the study of the Slice-Ribbon Conjecture.

Contribution

It introduces a universal method for calculating linking numbers via dihedral covers, applicable to all three-manifolds and relevant for knot obstructions.

Findings

Provides an explicit algorithm for linking number computation

Applicable to all closed, oriented three-manifolds

Supports research on the Slice-Ribbon Conjecture

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 $3$-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.