Detecting and visualizing 3-dimensional surgery

Stathis Antoniou, Louis H. Kauffman, Sofia Lambropoulou

Research output: Contribution to journalArticlepeer-review

Abstract

Topological surgery in dimension 3 is intrinsically connected with the classification of 3-manifolds and with patterns of natural phenomena. In this, mostly expository, paper, we present two different approaches for understanding and visualizing the process of 3-dimensional surgery. In the first approach, we view the process in terms of its effect on the fundamental group. Namely, we present how 3-dimensional surgery alters the fundamental group of the initial manifold and present ways to calculate the fundamental group of the resulting manifold. We also point out how the fundamental group can detect the topological complexity of non-trivial embeddings that produce knotting. The second approach can only be applied for standard embeddings. For such cases, we give new visualizations of 3-dimensional surgery as rotations of the decompactified 2-sphere. Each rotation produces a different decomposition of the 3-sphere which corresponds to a different visualization of the 4-dimensional process of 3-dimensional surgery.

Original languageEnglish
Article number1940015
Number of pages24
JournalJournal of Knot Theory and its Ramifications
Volume28
Issue number13
DOIs
Publication statusPublished - 1 Nov 2019

Keywords

  • 2-sphere
  • 3-manifold
  • 3-space
  • 3-sphere
  • blackboard framing
  • decompactification
  • framed surgery
  • fundamental group
  • handle
  • knot group
  • knots
  • Poincaré sphere
  • rotation
  • stereographic projection
  • surgery visualization
  • topological process
  • Topological surgery
  • topology change
  • torus
  • TOPOLOGICAL SURGERY
  • Poincare sphere

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