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Reidemeister's theorem using transversality

Hoel Queffelec

Abstract

We recover Reidemeister's theorem using smooth functions and transversality.

Reidemeister's theorem using transversality

Abstract

We recover Reidemeister's theorem using smooth functions and transversality.

Paper Structure

This paper contains 10 sections, 2 theorems, 25 equations, 6 figures.

Table of Contents

  1. Reidemeister's theorem
  2. Transversality
  3. Proving the theorem of Reidemeister
  4. Diagrams
  5. Isotopies
  6. First condition
  7. Second condition
  8. Third condition
  9. Anything else?
  10. Reidemeister-like theorems

Key Result

Theorem \oldthetheorem

Two diagrams represent the same knot if and only if they are related by a sequence of planar isotopies and of elementary moves of the following kind:

Figures (6)

  • Figure 1: Three knots: left and right-handed trefoil, trivial knot
  • Figure 2: Three diagrams for the trivial knot
  • Figure 3: Wikipedia illustration for the Reidemeister moves. Image credit: Parcly Taxel.
  • Figure 4: Curves in the plane
  • Figure 5: Local model for knot diagrams
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem \oldthetheorem: Reidemeister's theorem
  • Theorem \oldthetheorem: Multijet transversality theorem, Theorem 4.13 in GoGui