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Linking Numbers in Three-Manifolds.

Patricia Cahn1, Alexandra Kjuchukova2

  • 1Smith College, Northampton, USA.

Discrete & Computational Geometry
|November 1, 2021
PubMed
Summary
This summary is machine-generated.

This study presents an algorithm to calculate the linking number between curves in three-manifolds using dihedral covers. This method is applicable to all manifolds and aids in analyzing knot obstructions, potentially testing the Slice-Ribbon Conjecture.

Keywords:
3-manifoldsKnotsLinking numbers

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Area of Science:

  • Topology
  • Knot Theory
  • Geometric Topology

Background:

  • Three-manifolds and simple closed curves are fundamental in topology.
  • Linking numbers quantify the interweaving of curves in a manifold.
  • The Slice-Ribbon Conjecture remains a key open problem in knot theory.

Purpose of the Study:

  • To develop an explicit algorithm for computing linking numbers between rationally null-homologous curves in any closed, oriented three-manifold.
  • To establish a connection between linking number computation and the structure of dihedral covers of the knot complement.
  • To provide tools for evaluating ribbon obstructions and testing the Slice-Ribbon Conjecture.

Main Methods:

  • Representing a three-manifold M as an irregular dihedral three-fold cover of the knot complement S^3 \ K.
  • Utilizing presentations of the manifold derived from this cover structure.
  • Developing an explicit computational algorithm based on these presentations.

Main Results:

  • An explicit algorithm for computing the linking number between two curves K and L in M.
  • The algorithm's applicability to all closed, oriented three-manifolds.
  • Demonstration that linking numbers computed are essential for evaluating ribbon obstructions.

Conclusions:

  • The developed algorithm provides a universal method for linking number computation in three-manifolds.
  • This work offers a new perspective on knot obstructions and their relation to manifold structures.
  • The findings contribute to the ongoing efforts to test conjectures like the Slice-Ribbon Conjecture.