Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On continuous 2-frieze patterns

Serge Tabachnikov

Abstract

We define and study a continuous version of 2-frieze patterns, a combinatorial structure closely related with frieze patterns of Coxeter and Conway. We describe the relation of continuous 2-friezes with the moduli space of projective curves and relate the (pre)symplectic structure on the space of closed 2-friezes, considered as a cluster variety, with the Adler-Gelfand-Dikii bracket on the space of 3rd order differential operators.

On continuous 2-frieze patterns

Abstract

We define and study a continuous version of 2-frieze patterns, a combinatorial structure closely related with frieze patterns of Coxeter and Conway. We describe the relation of continuous 2-friezes with the moduli space of projective curves and relate the (pre)symplectic structure on the space of closed 2-friezes, considered as a cluster variety, with the Adler-Gelfand-Dikii bracket on the space of 3rd order differential operators.
Paper Structure (21 sections, 14 theorems, 82 equations)

This paper contains 21 sections, 14 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Motivation: continuous frieze patterns
  3. Continuous 2-friezes and continuous $SL_3$-tilings
  4. Continuos 2-friezes and projective curves
  5. Proof.
  6. Projective curves and differential operators.
  7. Proof.
  8. Continuous $SL_3$-tilings and 2-friezes from projective curves.
  9. Proof.
  10. Proof.
  11. Closed positive continuous 2-friezes.
  12. Proof.
  13. Proof.
  14. Self-dual closed continuous 2-friezes.
  15. Proof.
  16. ...and 6 more sections

Key Result

Theorem 1

All continuous frieze patterns are obtained from $SL_2$-equivalence classes of convex projective curves: $H(x,y)=\det(\Gamma(x),\Gamma(y)).$

Theorems & Definitions (20)

  • Definition 1
  • Theorem 1: OT
  • Theorem 2: OT
  • Definition 2
  • Definition 3
  • Lemma 4
  • Lemma 5
  • Proposition 6
  • Lemma 7
  • Proposition 8
  • ...and 10 more