Combinatorial Map Encodings

Updated 7 February 2026

Combinatorial map encodings are algebraic representations that use permutation pairs to capture the cyclic order of edges and faces in embedded graphs.
They enable systematic enumeration and canonicalization by reducing local embedding data to algorithmically tractable, invariant permutation formats.
Applications span topological graph theory, virtual knot theory, and tensor models, offering deep insights into surface maps and complex combinatorial structures.

Combinatorial map encodings provide a rigorous algebraic representation of graphs embedded on surfaces, capturing topological, combinatorial, and geometric information via permutations acting on finite sets. These encoding schemes form the backbone of modern research in topological graph theory, enumeration, virtual knot theory, random geometry, and algebraic geometry through their connections to surface maps, tensor models, and related combinatorial structures. The core principle is the reduction of discrete surface maps and cellular embeddings to purely local, permutation-based data that is amenable to algorithmic manipulation, enumeration, canonicalization, and algebraic analysis.

1. Foundations: Definitions and Core Permutation Models

A combinatorial (orientable) map is typically encoded as a triple or pair of permutations on a finite set of labels representing “darts,” “half-edges,” or “flags.” The standard formalism is as follows:

Given a finite set $H$ of $2m$ (or $4N$ in the case of 4-regular maps) darts,
A fixed-point-free involution $\alpha\in S_H$ pairs darts into edges,
A permutation $\sigma\in S_H$ (with cycles of prescribed length) encodes the cyclic order of darts around each vertex,
The subgroup $\langle \alpha, \sigma \rangle$ acts transitively on $H$ (connectedness),
The faces of the embedding are encoded by the cycles of $\varphi := \sigma\alpha$ .

This encoding is isomorphic to other common models, such as the Tutte triplet $(\sigma, \alpha, \varphi)$ , rotation systems, or the triple $(B, \sigma, \alpha)$ on the set of “flags” (Cori et al., 2022). The permutation model supports a range of encoding variants, including those for orientable and non-orientable surfaces, one-face or multi-face maps, regular or general degree, and more specialized contexts such as toroidal 4-regular projections in virtual knot theory (Omelchenko, 21 Jan 2026).

2. Canonicalization, Normalization, and Invariants

To enumerate map types up to isomorphism, duplicate-free canonical representatives must be selected. The main strategies are:

Unsensed Canonical Representative: Two encodings $(\alpha_1,\sigma_1)$ and $(\alpha_2,\sigma_2)$ are equivalent iff there exists $\pi\in S_H$ such that $(\alpha_2,\sigma_2) = (\pi \alpha_1 \pi^{-1}, \pi \sigma_1 \pi^{-1})$ , with an additional check for unsensed symmetry via $(\alpha_2, \sigma_2^{-1})$ (Omelchenko, 21 Jan 2026).
The canonical form is typically the lex-minimal element under all relabelings and inversion, often determined via a deterministic breadth-first traversal and normalization of labels.
Knot Normalization: In Zeps’s combinatorial map encoding, the “normalized knot” method selects labels so that the knot $\mu = \pi \rho$ becomes a canonical cycle form, resulting in significant computational and notational reduction. Parity of labeled corners directly identifies features (cut/cycle edges), streamlining many topological operations (Zeps, 2010).

This systematic relabeling is crucial to creating bijections between combinatorial objects and ensuring complete invariance under map isomorphism, particularly when generating exhaustive tables of maps of given genus and size.

3. Specialized Encodings: Unicellular Maps, Double Occurrence Words, and Chord Diagrams

For unicellular (one-face) maps and related structures, specialized combinatorial encodings enable fine-grained analysis:

Unicellular Map Encodings: Represented by $(\alpha, \sigma)$ with the requirement that the face permutation $\gamma = \alpha \circ \sigma$ is a single cycle (Chapuy, 2010). Chapuy’s bijection iteratively decomposes or constructs such maps using cycle gluing/slicing operations indexed by trisections, giving an explicit algorithmic bridge to plane trees with distinguished vertices. Formulas such as

$2g\epsilon_g(n) = \sum_{p=0}^{g-1} \binom{n+1-2p}{2(g-p)+1} \epsilon_p(n)$

underlie enumerative results, with vertex and edge permutations serving as map encoding (Chapuy, 2010).

Double Occurrence Word and Chord Diagram Encoding: A rooted combinatorial map (with a distinguished dart) and a chosen spanning quasi-tree can be encoded as a double occurrence word: a sequence where each edge label appears exactly twice, corresponding to basepointed traversals of the tour permutation $\tau_S$ $τ_{S}$ (Cori et al., 2022). This encoding is in bijection with:
- Rooted bicolored chord diagrams (where edges in the quasi-tree are blue chords, others red)—in particular, quotienting out the basepoint and color preserves the full mapping structure.
- Ordered matchings, providing an alternate viewpoint especially useful for enumerative problems involving loopless maps.

Such encodings facilitate not only combinatorial enumeration but also bijective proofs and deeper algebraic analysis, including explicit connections to circle graphs, fundamental interlacement graphs, and the structure of dual maps.

4. Advanced Structures: Stuffed Walsh Maps, Colored Triangulations, and Tensor Models

Higher-dimensional and colored triangulations of pseudo-manifolds admit explicit combinatorial map encodings via edge-colored graphs and their associated map models:

Stuffed Walsh Maps: Any family of colored graphs encoding triangulations built from fixed boundary patterns (bubbles) can be mapped bijectively to “stuffed Walsh maps,” a generalization of bipartite Walsh maps in which hyperedges are replaced by fixed colored submaps. The black vertices of the stuffed Walsh map arise from cycles of color-$0$ edges coupled with prescribed bubble pairings, while the blue submaps encode the internal combinatorial structure of the bubbles (Bonzom et al., 2015).
Matrix and Tensor Model Correspondence: These map encodings form the combinatorial backbone of Feynman diagrams in colored tensor models, with explicit mapping between colored-graph (triangulation) expansions and multi-matrix model expansions via the stuffed Walsh map formalism (Bonzom et al., 2015).

This framework preserves bijective correspondence in both directions and is extensible to higher dimensions, arbitrary boundary conditions, and nontrivial topologies.

5. Combinatorial Map Encodings in Virtual Knot Theory

A significant recent application is the generation and classification of virtual knots and links on thickened surfaces, notably the torus $T^2$ :

Permutation Pair Model for Genus-One Projections: Any cellular 4-regular map (projection) on the torus is encoded as a pair $(\alpha, \sigma)$ $(α, σ)$ of permutations on $H = \{1, 2, ..., 4N\}$ $H = {1, 2, ..., 4 N}$ , with $\alpha$ $α$ a fixed-point-free involution and $\sigma$ $σ$ the vertex rotation (each a 4-cycle) (Omelchenko, 21 Jan 2026). The face permutation $\varphi := \sigma\alpha$ $φ := σ α$ determines the surface genus, and filtering for no monogons, looplessness, and torus-topology yields only valid projections. Additional combinatorial assignments:
- Crossing information is encoded as an $N$ -bit vector.
- Immediate Reidemeister II reducibility (bigons) is a local, purely bitwise criterion on the permutations and bit assignment.
- The Kauffman bracket variant for $T^2$ is computed as a complete state sum within the permutation formalism, with cycles of appropriately defined auxiliary permutations in bijection with state-circles (distinguished homologically via binary incidence vectors) (Omelchenko, 21 Jan 2026).

This purely combinatorial approach, requiring no geometric representation, has enabled the first canonical, completely reproducible tabulation of all genus-one virtual knots up to $N=8$ crossings, bridging computational knot theory and map enumeration.

6. Applications, Extensions, and Algorithmic Implications

Combinatorial map encodings, by reducing arbitrary surface-embedded structures to permutation data, readily support:

Efficient enumeration pipelines: exhaustive generation, isomorphism testing, duplicate elimination, and property extraction,
Integration with local search and optimization methodologies where the encoding induces a smoother, often enriched landscape in the space of “genotypes,” facilitating more effective exploration of large search spaces in combinatorial optimization (Klemm et al., 2011),
Generalization to colored and higher-dimensional structures underpinning random tensor models, topological recursion, and quantum gravity discretizations (Bonzom et al., 2015),
Algorithmic computation of surface invariants, prime decomposition, and detection of local or global reductions (e.g., Reidemeister moves, trisection slicing),
The design of concise data structures and renumbering schemes (e.g., the normalized knot) that optimize both theoretical tractability and computation.

A plausible implication is the further exploitation of combinatorial map encoding frameworks in both classical and emerging algorithmic contexts where structural symmetry, redundancy elimination, and combinatorial invariants are essential.

7. Comparative Overview and Directions

A comparative synopsis of prominent combinatorial map encoding schemes is summarized below:

Encoding Paradigm	Key Elements	Main Application Contexts
$(\alpha, \sigma)$ permutation pair	Edges as involution, vertex order as cycles	General orientable maps, torus projections
Knot normalization (Zeps)	Canonicalized permutation, parity labeling	Concise maps, fast edge/cycle identification
Double occurrence/chord word	Edge traversal as word, chord/bicolored diagram	Enumeration, loopless/planar, DFS-trees
Stuffed Walsh map	Bipartite map with colored submaps/bubbles	Colored triangulations, tensor models

The diversity and adaptability of these encoding strategies continue to fuel advances in enumerative combinatorics, algorithmic topology, mathematical physics, and computational geometry, suggesting broad applicability to new domains requiring rigorous and efficient representations of embedded discrete structures.