Doodle Invariants in Planar and Virtual Topology

• Doodle invariants are algebraic objects assigned to immersed circles with transverse double points, remaining unchanged under isotopies and R1/R2 moves.
• They encompass a range of constructions—from Alexander-type polynomial invariants to complete finite-type and chord-diagram series—that capture subtle structural differences.
• Twin-group frameworks and canonical combinatorial encodings underpin their classification in both planar and virtual settings, linking representation theory with low-dimensional topology.

Doodle invariants are quantities or algebraic objects assigned to doodles—immersions of disjoint unions of circles with only transverse double points and no triple points—such that they are unchanged under the equivalence relation of doodle theory. In the classical planar setting, equivalence is generated by isotopy of $S^2$ together with $R_1$ and $R_2$, but not $R_3$; on surfaces, stable equivalence also allows surgeries away from the diagram. The absence of $R_3$ makes doodle theory structurally different from knot theory: minimal representatives are unique in the $S^2$ setting, complete combinatorial invariants exist, and invariant theories range from polynomial and finite-type constructions to Gauss-code, coloring, genus, bordism, and group-theoretic frameworks (Cisneros et al., 2020, Mostovoy, 2024, Bartholomew et al., 2016).

1. Foundational setting and invariant-theoretic consequences

A planar doodle is an immersion

$D:\bigsqcup_n S^1 \longrightarrow S^2$

with only transverse double points and no triple or higher intersections. An oriented doodle is one in which each circle is oriented. Two planar doodles are equivalent if they are related by a homotopy through immersions with no triple points; on diagrams this is equivalent to isotopies of the sphere together with planar $R_1$ and $R_2$, but not $R_3$. A doodle with more than one component is unlinked, or split, when it decomposes into doodles lying in disjoint disks on $R_1$0 (Cisneros et al., 2020).

On closed surfaces, the same immersion condition is used, but equivalence is enlarged by homeomorphic equivalence, flat $R_1$1 and $R_1$2 moves, and surface surgery away from the diagram via handle addition and elimination. This produces doodles on arbitrary closed oriented surfaces; planar doodles are precisely those admitting a representative on $R_1$3. The same work establishes a natural one-to-one correspondence between doodles on surfaces and virtual doodles on the plane (Bartholomew et al., 2016).

A common misconception is that doodles are merely knot or link diagrams without over/under data. The decisive difference is the prohibition of $R_1$4. In the planar theory this yields a unique minimal-crossing representative in each equivalence class, and in the surface theory it yields a unique minimal diagram up to homeomorphism of the supporting surface (Cisneros et al., 2020, Bartholomew et al., 2016).

2. Canonical representatives and combinatorial encodings

The existence of canonical minimal representatives makes several combinatorial encodings genuinely invariant. For doodles on $R_1$5, arrow diagrams encode crossings by oriented chords on the parameter circle; a doodle is determined up to isotopy by its arrow diagram, and minimality is reflected by the absence of reducible $R_1$6 and $R_1$7 patterns in that diagrammatic language (Mostovoy, 2024).

For oriented virtual doodles, the most explicit canonical form is the left canonical Gauss code. A Gauss code records the order in which left and right branches of real crossings are encountered along the oriented circle. After normalization to a left preferred form and minimization in the orbit under cyclic shift, one obtains a unique left canonical representative. For doodles, one first passes to the unique minimal virtual diagram and then takes its canonical code. The resulting assignment $R_1$8 is a complete invariant of oriented virtual doodles, and the analogous unoriented version is complete after quotienting by reversal (Bartholomew et al., 2018).

Planar doodles also admit a coarser but useful encoding by region counts. If a minimal doodle diagram has $R_1$9 crossings and $R_2$0 regions that are $R_2$1-gons, then

$R_2$2

The tuple $R_2$3 is the doodle code. It is only a partial characterization, because different doodles can share the same region counts, but it strongly constrains possibilities and is central to enumeration via dual graphs (Bartholomew et al., 2023).

The same graph-theoretic framework yields invariant connectivity notions. A prime doodle is 3-connected; a super prime doodle is 4-connected. For prime doodles, every pair of regions are either disjoint, meet in a single vertex, or meet in a single edge. For super prime doodles, Tutte’s theorem implies that the underlying planar graph has a Hamiltonian circuit, and this leads to Hamiltonian codes that refine doodle codes in the super prime regime (Bartholomew et al., 2023).

3. Alexander-type polynomial invariants

The first classical-looking polynomial invariant for oriented planar doodles is the Alexander-type invariant $R_2$4. Its construction follows the braid-to-polynomial paradigm from knot theory, but with twins in place of braids. The twin group

$R_2$5

is a right-angled Coxeter group whose elements are planar braids on $R_2$6 strands. Khovanov’s Alexander-type theorem states that every doodle is the closure of some twin, and Gotin’s Markov-type theorem describes when two twins have equivalent closures (Cisneros et al., 2020, Gotin, 2018).

The invariant is built from a two-parameter deformation of the Tits representation,

$R_2$7

specialized later to $R_2$8. The normalization depends on determinants of words $R_2$9, and these determinants satisfy

$R_3$0

where $R_3$1 are Chebyshev polynomials of the second kind. For $R_3$2,

$R_3$3

and the doodle invariant is the monic generator of the ideal

$R_3$4

This produces $R_3$5, an invariant of oriented doodles (Cisneros et al., 2020).

Its behavior closely parallels the Alexander polynomial. At the $R_3$6-level there is a skein-type relation, and $R_3$7 vanishes on unlinked multi-component doodles, directly mirroring the vanishing of the classical Alexander polynomial on split links with more than one component. The $R_3$8 are only defined up to multiplication by even powers of $R_3$9 under Markov moves, and $R_3$0 removes that ambiguity by taking the monic generator. The paper emphasizes that the skein relation does not in general survive the passage from $R_3$1 to $R_3$2 (Cisneros et al., 2020).

The invariant is nontrivial in low-complexity examples. The trivial doodle has $R_3$3. The first nontrivial one-component doodle, the 4-poppy, represented by $R_3$4, has

$R_3$5

The Borromean doodle, represented by $R_3$6, has

$R_3$7

For the family $R_3$8 given by the closures of $R_3$9, the invariant distinguishes different $S^2$0 (Cisneros et al., 2020).

4. Finite-type, complete series, and chord-diagram invariants

Finite-type theory for doodles is unusually strong. One cited result states that Vassiliev invariants classify doodles, a statement not known for knots, and this already situates doodles as an exceptional case among low-dimensional diagram categories (Cisneros et al., 2020).

A complete diagrammatic realization of this principle is given by the invariant $S^2$1 for doodles on $S^2$2. Starting from the vector space $S^2$3 spanned by arrow diagrams, one defines

$S^2$4

the sum over all sub-arrow-diagrams obtained by deleting arbitrary subsets of chords. Quotienting by the relations induced from $S^2$5 and $S^2$6 yields a target $S^2$7, then truncations $S^2$8, and hence invariants

$S^2$9

The main theorem states that $D:\bigsqcup_n S^1 \longrightarrow S^2$0 is a finite type invariant of doodles of order at most $D:\bigsqcup_n S^1 \longrightarrow S^2$1; if $D:\bigsqcup_n S^1 \longrightarrow S^2$2 is a non-trivial doodle with $D:\bigsqcup_n S^1 \longrightarrow S^2$3 or fewer crossings, then $D:\bigsqcup_n S^1 \longrightarrow S^2$4; and if $D:\bigsqcup_n S^1 \longrightarrow S^2$5 are non-equivalent doodles with $D:\bigsqcup_n S^1 \longrightarrow S^2$6 or fewer crossings, then $D:\bigsqcup_n S^1 \longrightarrow S^2$7. Consequently, the inverse limit $D:\bigsqcup_n S^1 \longrightarrow S^2$8 is a complete invariant, and its coefficients at degree-$D:\bigsqcup_n S^1 \longrightarrow S^2$9 diagrams are finite type invariants of order at most $R_1$0 (Mostovoy, 2024).

This construction is close in spirit to Polyak–Viro subdiagram expansions, but the combinatorics of doodles make the result stronger than in the knot case: the universal series is actually injective. The proof passes through quiver diagrams, reduction of adjacent chords, and a basis theorem for reduced diagrams, allowing one to detect a minimal doodle by a distinguished basis element appearing with coefficient $R_1$1 (Mostovoy, 2024).

A different chord-diagram interaction appears in the 2025 representation of Milnor’s triple linking number. For a 3-component doodle $R_1$2, Hirata uses a doodle invariant

$R_1$3

defined by smoothing $R_1$4, counting signed intersections of $R_1$5 and $R_1$6 inside the bounded regions, and correcting by the orientation of each smoothed component. This $R_1$7-invariant is integer-valued, antisymmetric under permutation of components, changes by $R_1$8 under the forbidden $R_1$9-type move, and appears in the formula

$R_2$0

Here the additional terms are signed counts of intersections in generalized chord diagrams associated to the link components. This places doodle invariants directly into the degree-two structure of Milnor’s triple linking number (Hirata, 5 Oct 2025).

5. Virtual, surface, algebraic, and bordism invariants

Beyond planar polynomial and finite-type invariants, doodle theory has a broad surface and virtual sector with several distinct invariant packages.

For doodles on surfaces, the unique minimal diagram determines the minimal crossing number and the genus of the doodle; in the virtual formulation, the virtual area number equals that genus. The same framework classifies planar doodles up to eight crossings and gives explicit families such as the generalized Borromean doodles $R_2$1, the gyro family $R_2$2, and the ortho family $R_2$3, distinguished by component counts and minimal-diagram combinatorics (Bartholomew et al., 2016).

For virtual doodles, doodle switches provide coloring invariants. A doodle switch is a set with a binary operation satisfying the axioms in which $R_2$4 if and only if $R_2$5, right multiplication is bijective, and the map $R_2$6 is bijective. The fundamental doodle switch $R_2$7 is defined from semiarcs and crossing relations, and $R_2$8 is invariant under virtual doodle equivalence. The doubled version $R_2$9 yields doubled coloring numbers $R_3$0, and in the examples computed with a 4-element doodle switch, $R_3$1 distinguishes some diagrams that $R_3$2 does not (Bartholomew et al., 2018).

For one-component virtual doodles, the skew-symmetric augmented matrix

$R_3$3

is built from homology intersection numbers of primitive curves on the Carter surface. Up to permutations and elementary extensions/reductions, its $R_3$4-equivalence class is invariant under stable $R_3$5-equivalence. A one-component doodle diagram is almost classical if and only if all the intersection numbers vanish, equivalently if $R_3$6 is the zero matrix; classical doodles have trivial matrix class. The Kishino doodle yields an irreducible nontrivial augmented matrix and is therefore non-classical (Ocampo et al., 2024).

A different bordism-oriented theory appears for 2-moderate immersions in $R_3$7 or $R_3$8. For embedded oriented doodles in the strip, tangency counts define invariants $R_3$9, with $R_1$00, and

$R_1$01

via $R_1$02. For immersed oriented doodles there is an exact sequence

$R_1$03

where $R_1$04 counts signed differences of crossing types, and the kernel $R_1$05 contains a subgroup $R_1$06 generated by explicit families built from “$R_1$07” and “8” doodles (Katz, 2023).

6. Twin groups, Alexander–Markov correspondences, and extended invariant frameworks

The twin-group viewpoint is the main bridge from doodles to representation theory. On $R_1$08, Gotin’s Markov theorem identifies doodles with twin closures modulo Markov moves, so any family of functions on twin groups invariant under those moves descends to a doodle invariant. This is exactly the mechanism used by the Alexander-type invariant $R_1$09, and it is the doodle analogue of the braid–Markov framework for classical link invariants (Gotin, 2018, Cisneros et al., 2020).

This correspondence has now been extended to closed surfaces, including non-orientable ones. Twisted virtual doodles are stable equivalence classes of doodles on closed surfaces that may be non-orientable; their planar representatives carry real crossings, virtual crossings, and bars. The parity of the total number of bars is invariant under the extended Reidemeister system, so odd-bar diagrams cannot be equivalent to virtual doodles. To capture these objects braid-theoretically, the twisted virtual twin group $R_1$10 is introduced with generators $R_1$11, $R_1$12, and $R_1$13, combining classical twin, virtual, and bar relations. Every twisted virtual doodle is the closure of a twisted virtual twin, and two such twins have equivalent closures if and only if they are related by the twisted Markov moves TM0–TM3 (Negi et al., 12 Nov 2025).

The same work supplies a large algebraic environment for future invariant theory. The pure twisted virtual twin group $R_1$14 admits an explicit presentation; $R_1$15; $R_1$16 decomposes both as an iterated semidirect product of free products $R_1$17 and as $R_1$18, where $R_1$19 is an irreducible right-angled Artin group. From these decompositions it follows that $R_1$20 and $R_1$21 have trivial center and are residually finite as well as Hopfian. This does not itself produce a new numerical doodle invariant, but it provides the exact Markov-compatible algebraic setting in which such invariants can be constructed (Negi et al., 12 Nov 2025).

Taken together, these developments show that doodle invariants now occupy several distinct but interacting layers: canonical combinatorial representatives, complete diagrammatic series, Alexander-type polynomials, coloring and genus invariants, homology-intersection obstructions to classicality, bordism invariants in ruled surfaces, and twin-group frameworks on both orientable and non-orientable surfaces. The cumulative picture is unusual in low-dimensional topology: doodles admit both complete combinatorial invariants and classical-looking representation-theoretic constructions, while still supporting substantial open problems about how these theories interact (Mostovoy, 2024, Cisneros et al., 2020, Negi et al., 12 Nov 2025).