Document Type
Article
Publication Date
9-1-2021
Publication Title
Discrete and Computational Geometry
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 S3 branched along a knot α⊂ S3. 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 α can be derived from dihedral covers of α. 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.
Keywords
3-manifolds, Knots, Linking numbers
Volume
66
Issue
2
First Page
435
Last Page
463
DOI
10.1007/s00454-021-00287-3
ISSN
01795376
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Rights
Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy.
Recommended Citation
Cahn, Patricia and Kjuchukova, Alexandra, "Linking Numbers in Three-Manifolds" (2021). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/113
Comments
Peer reviewed accepted manuscript.