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 $2n$.

• 05C10 - Planar graphs; geometric and topological aspects of graph theory
• 57K12 - Generalized knots (virtual knots, welded knots, quandles, etc.)
• 57K16 - Finite-type and quantum invariants, topological quantum field theories (TQFT)