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A complete invariant for doodles on a 2-sphere

Jacob Mostovoy

Abstract

We define a complete invariant for doodles on a 2-sphere which takes values in series of chord diagrams of certain type. The coefficients at the diagrams with $n$ chords are finite type invariants of doodles of order at most $2n$.

A complete invariant for doodles on a 2-sphere

Abstract

We define a complete invariant for doodles on a 2-sphere which takes values in series of chord diagrams of certain type. The coefficients at the diagrams with chords are finite type invariants of doodles of order at most .
Paper Structure (5 sections, 18 equations, 6 figures)

This paper contains 5 sections, 18 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Doodles and their arrow diagrams
  3. Quiver diagrams and the diagram invariants
  4. The diagram invariants are of finite type
  5. Some remarks

Figures (6)

  • Figure 1: A doodle and its arrow diagram
  • Figure 2: Reduction of a chord diagram
  • Figure 3: Reduction of a quiver diagram
  • Figure 4: A pair of intersecting adjacent chords pointing in the same direction and what they might look like in a doodle. The points $B$ and $C$ must be joined in $S^2$ by a segment of the doodle containing no other intersection points, which is impossible.
  • Figure 5: Doodles whose arrow diagrams have adjacent chords
  • ...and 1 more figures

Theorems & Definitions (4)

  • proof
  • proof
  • proof
  • proof : Sketch of the proof of Proposition \ref{['propft']}