Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Legendrian Hopf links

Hansjörg Geiges, Sinem Onaran

Abstract

We completely classify Legendrian realisations of the Hopf link, up to coarse equivalence, in the 3-sphere with any contact structure.

Legendrian Hopf links

Abstract

We completely classify Legendrian realisations of the Hopf link, up to coarse equivalence, in the 3-sphere with any contact structure.

Paper Structure

This paper contains 40 sections, 15 theorems, 101 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contact structures on $S^3$
  3. The link complement
  4. Hopf links in $(S^3,\xi_{\mathrm{st}})$
  5. Strongly exceptional Hopf links
  6. Kirby moves
  7. Computing the invariants
  8. Thurston--Bennequin invariant
  9. Rotation number
  10. The $d_3$-invariant
  11. Detecting exceptional links
  12. Exceptional unknots
  13. Legendrian realisations
  14. Case (b1)
  15. Case (b2)
  16. ...and 25 more sections

Key Result

Theorem 1.2

Up to coarse equivalence, the Legendrian realisations of the Hopf link are as follows. In all cases, the classical invariants determine the Legendrian realisation.

Figures (18)

  • Figure 1: Legendrian Hopf link in $(S^3,\xi_{\mathrm{st}})$.
  • Figure 2: Kirby diagram of a Hopf link I.
  • Figure 3: Kirby diagram of a Hopf link II.
  • Figure 4: Kirby diagram of a Hopf link III.
  • Figure 5: Kirby diagram of a Hopf link IV.
  • ...and 13 more figures

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem 1.2
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Remark 3.3
  • Lemma 5.1
  • ...and 22 more