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Six proofs of the Fáry--Milnor theorem

Anton Petrunin, Stephan Stadler

Abstract

We sketch several proofs of Fáry--Milnor theorem.

Six proofs of the Fáry--Milnor theorem

Abstract

We sketch several proofs of Fáry--Milnor theorem.
Paper Structure (6 sections, 7 theorems, 26 equations, 13 figures)

This paper contains 6 sections, 7 theorems, 26 equations, 13 figures.

Table of Contents

  1. Milnor--Fenner.
  2. Fáry.
  3. Pannwitz--Hopf--Schmitz--Denne.
  4. Alexander--Bishop.
  5. Ekholm--White--Wienholtz.
  6. Cantarella--Kuperberg--Kusner--Sullivan.

Key Result

Proposition 1

Assume that a height function $(x,y,z)\to z$ has only one local maximum on a simple closed polygonal curve $\alpha$ and all the vertices of the polygonal curve are at different heights. Then $\alpha$ is a trivial knot.

Figures (13)

  • Figure 1: Trefoil knot.
  • Figure 2: Triangular isotopy.
  • Figure 3: Spherical slice.
  • Figure 4: Definitions of angles.
  • Figure 5: Projection of an unknot.
  • ...and 8 more figures

Theorems & Definitions (20)

  • Proposition 1
  • proof
  • proof : Proof of the main theorem
  • proof
  • Proposition 3
  • proof
  • proof : Proof of the main theorem
  • Proposition 4
  • proof : Proof of the main theorem modulo Proposition \ref{['prop:quadrisecant']}
  • Lemma 5
  • ...and 10 more