Monodromy Graphs: Theory & Applications

• Monodromy graphs are graph-theoretic devices used to encode monodromy actions, permutation groups, and degeneration data across algebraic and geometric frameworks.
• They represent local continuation data with vertices indexing fibers or states and edges encoding homotopy paths, enabling applications in polynomial system solving and Hurwitz theory.
• Their structured combinatorial nature underpins computational methods and certification techniques in diverse domains such as moduli spaces, singularity resolution, and group theoretic constructions.

Monodromy graphs are graph-theoretic devices used to encode monodromy actions, monodromy groups, or monodromy-related combinatorics arising from deformation, analytic continuation, degeneration, and covering theory. The term is not univocal. In numerical algebraic geometry it denotes a decorated graph whose vertices are polynomial systems and whose edges are homotopies between parameter values; in Hurwitz theory it denotes a weighted graph over an interval encoding branched covers and factorisations; in logarithmic and singularity theory it is often essentially the dual graph or resolution graph of a degeneration; in surface topology it may be a metric ribbon graph or filtered graph realising a mapping class; and in recent algebraic graph theory it denotes a double-coset construction $\mathcal{M}(G;U,\rho,\tau)$ representing arbitrary graphs. What these usages share is the conversion of continuation data into finite combinatorics (Duff et al., 2016, Hahn, 2017, Achinger et al., 2018, Yuan et al., 22 Sep 2025).

1. Scope and recurring structure

Across the literature, monodromy graphs organise how local data are transported globally. In the polynomial-system setting, the transport concerns solutions in a fiber over parameter space. In Hurwitz theory, it concerns sheets of a branched cover or cycles in a symmetric-group factorisation. In semistable or logarithmic degeneration, it concerns vanishing cycles, residues, and nearby cycles. In mapping-class and singularity theory, it concerns periodic or pseudo-periodic automorphisms of surfaces. In algebraic graph theory, the graph itself is reconstructed from group-theoretic monodromy data.

Setting	Graph data	Monodromy role
Parametrized polynomial systems	Vertices $F_{p_i}$, decorated edges $(\gamma_1,\gamma_2)$	Tracks solution correspondences
Hurwitz theory	Weighted graph over $0,\dots,b+1$	Encodes cut-and-join and ramification
Log degenerations	Dual graph of components and nodes	Encodes residues, nearby cycles, Picard-Lefschetz data
Algebraic graph theory	Double cosets $Ug\langle\rho\rangle$	Represents a graph via group action

A recurrent pattern is that vertices index fibers, components, states, or cosets, while edges encode continuation paths, gluings, incidences, or generators. This suggests a family resemblance rather than a single canonical definition. A common misconception is therefore that “monodromy graph” always refers to one specific object; the sources instead use the phrase for several formally distinct but conceptually aligned constructions (Duff et al., 2016, Achinger et al., 2018, Yuan et al., 22 Sep 2025).

2. Decorated graphs for parametrized polynomial systems

For a family of polynomial systems $F_p(x)=0$, the solution variety

$V=\{(F_p,x)\mid F_p(x)=0\}\to B$

is a branched covering map whose generic fiber has $d$ points. In this setting, the monodromy group is the set of permutations of the fiber induced by transporting solutions along loops in parameter space avoiding the discriminant. The graph-based framework of monodromy solvers represents this process by a finite undirected edge-decorated graph $G$: each vertex corresponds to a base system $F_{p_v}$, each edge between $F_{p_i}$0 and $F_{p_i}$1 is decorated by two scalars $F_{p_i}$2, and the edge corresponds to a homotopy given by a convex combination of $F_{p_i}$3. Self-edges are omitted because they induce trivial correspondences. Algorithmically, one starts from a seed system and solution, tracks known solutions along selected edges, updates correspondences between adjacent fibers, and continues until the expected solution count is reached, no new points can be found, or a stabilization heuristic is met. Static graphs such as flower or complete graphs may be fixed in advance, whereas dynamic graphs are augmented on the fly when progress stalls (Duff et al., 2016).

Under the idealised assumption that monodromy actions induced by random graph cycles behave as independent, uniformly distributed permutations of the $F_{p_i}$4-element fiber, the expected computational complexity becomes linear in $F_{p_i}$5. The corresponding probabilistic statement is

$F_{p_i}$6

where $F_{p_i}$7 is the minimal number of random permutations needed for transitivity, and $F_{p_i}$8 as $F_{p_i}$9. For a static graph with $(\gamma_1,\gamma_2)$0 edges, the maximum number of tracked homotopy paths is $(\gamma_1,\gamma_2)$1, whereas a naive coupon-collector strategy requires about $(\gamma_1,\gamma_2)$2 steps. The implementation in the MonodromySolver package in Macaulay2, built atop NumericalAlgebraicGeometry, supports random or potential-guided edge selection through heuristics such as potentialLowerBound and potentialE, and numerical solutions may be certified a posteriori via Smale’s alpha-theory when the system is regular and square with known solution count (Duff et al., 2016).

A certified extension replaces heuristic path tracking by interval-validated transport on a homotopy graph $(\gamma_1,\gamma_2)$3 embedded in parameter space. Vertices are generic parameter points $(\gamma_1,\gamma_2)$4, edges carry parameter paths $(\gamma_1,\gamma_2)$5, and the requirement is that for all $(\gamma_1,\gamma_2)$6 the system has $(\gamma_1,\gamma_2)$7 isolated solutions. Edge correspondences are then certified by interval arithmetic and the Krawczyk operator

$(\gamma_1,\gamma_2)$8

After saturation of the graph, a spanning tree $(\gamma_1,\gamma_2)$9 yields permutations from non-tree edges via

$0,\dots,b+1$0

The correctness theorem states that the resulting generators are actual monodromy permutations for the discovered solutions, and if $0,\dots,b+1$1 then the full group is obtained. Reported certified examples include a degree-23 Belyi polynomial with monodromy group $0,\dots,b+1$2, a symmetric cubic surface with monodromy group $0,\dots,b+1$3, the P3P problem with monodromy group $0,\dots,b+1$4, the 5-point pose problem with $0,\dots,b+1$5, and a nearest-point problem with monodromy group $0,\dots,b+1$6 (Duff et al., 18 Mar 2026).

3. Hurwitz theory, dessins, and graph representations of coverings

In Hurwitz theory, a monodromy graph of type $0,\dots,b+1$7 is a connected graph mapped to an interval with vertices over $0,\dots,b+1$8, where $0,\dots,b+1$9. Leaves over $Ug\langle\rho\rangle$0 are labelled by $Ug\langle\rho\rangle$1, leaves over $Ug\langle\rho\rangle$2 by $Ug\langle\rho\rangle$3, edges are weighted, and at every inner vertex the sums of incoming and outgoing edge weights agree. This graph is constructed from a symmetric-group factorisation $Ug\langle\rho\rangle$4 by successive cut-and-join operations associated with the transpositions $Ug\langle\rho\rangle$5. Simple double Hurwitz numbers are then computed as sums over monodromy graphs with multiplicity

$Ug\langle\rho\rangle$6

The same framework is refined for monotone, strictly monotone, and triply interpolated Hurwitz numbers by adding colorings, counters, and chain-path orderings, and this refinement supports algorithms for piecewise polynomiality and wall-crossing based on Ehrhart theory (Hahn, 2017).

The graph-theoretic meaning here is literal: fixing a combinatorial graph type reduces counting covers to counting lattice points in associated polytopes. In genus $Ug\langle\rho\rangle$7, the chamber polynomial for a fixed chamber is obtained by enumerating reduced monodromy graphs, assigning valid counter data along chain-paths, multiplying by internal edge weights, and summing. In arbitrary genus, the procedure yields quasi-polynomials after refining the chamber by partial orderings. The wall-crossing formulas are likewise expressed by cutting monodromy graphs along distinguished edges and reassembling smaller Hurwitz-type counts (Hahn, 2017).

A closely related usage appears in the theory of coverings of complex curves. For indecomposable coverings $Ug\langle\rho\rangle$8, the monodromy group is the Galois group of the Galois closure, or equivalently the image of the topological monodromy representation of $Ug\langle\rho\rangle$9. The associated monodromy graph, or dessin d’enfant, is described as a bipartite graph on the Riemann surface whose local valencies record ramification partitions. In the bounded-genus classification for sufficiently large degree, the generic case has monodromy group $F_p(x)=0$0 or $F_p(x)=0$1, while the non-$F_p(x)=0$2, non-$F_p(x)=0$3 cases are confined to explicit exceptional families; in these cases the ramification data and the group determine the possible dessins up to the classified possibilities (Neftin et al., 2024).

A different but structurally parallel development defines monodromy graphs via double cosets. Let $F_p(x)=0$4 with $F_p(x)=0$5, let $F_p(x)=0$6 be core-free, and let $F_p(x)=0$7. The monodromy graph $F_p(x)=0$8 has vertex set

$F_p(x)=0$9

and edge set

$V=\{(F_p,x)\mid F_p(x)=0\}\to B$0

with multiplicities defined through the relevant double-coset fibres. The main theorem states that every graph is isomorphic to such a monodromy graph, and setting $V=\{(F_p,x)\mid F_p(x)=0\}\to B$1 yields an arc-transitive graph. The same formalism underlies an enumeration theorem for orientable maps with a given monodromy group: if $V=\{(F_p,x)\mid F_p(x)=0\}\to B$2 admits $V=\{(F_p,x)\mid F_p(x)=0\}\to B$3 non-isomorphic regular maps and has $V=\{(F_p,x)\mid F_p(x)=0\}\to B$4 conjugacy classes of core-free subgroups, then the number of non-isomorphic maps with monodromy group $V=\{(F_p,x)\mid F_p(x)=0\}\to B$5 is $V=\{(F_p,x)\mid F_p(x)=0\}\to B$6 (Yuan et al., 22 Sep 2025).

4. Moduli spaces, local systems, and vanishing lattices

For meromorphic projective structures on a marked bordered surface $V=\{(F_p,x)\mid F_p(x)=0\}\to B$7, the monodromy map

$V=\{(F_p,x)\mid F_p(x)=0\}\to B$8

sends a signed meromorphic projective structure without apparent singularities to a non-degenerate framed $V=\{(F_p,x)\mid F_p(x)=0\}\to B$9 local system. The image lies in the union of domains of the cluster charts indexed by tagged triangulations, and the exchange graph of tagged triangulations organises these charts through flips. One source explicitly notes that it does not construct explicit “monodromy graphs”; instead, the behavior of the monodromy map is encoded combinatorially by the exchange graph and the adjacency of cluster charts. For poles of order $d$0, the image is determined precisely: it is the set $d$1 of non-degenerate framed representations. Fock–Goncharov coordinates attach cross-ratios

$d$2

to arcs of an ideal triangulation, and these coordinates supply the graph-theoretic parametrisation of the decorated monodromy data (Allegretti et al., 2018, Gupta et al., 2019).

In the Hitchin system for $d$3 and $d$4-Higgs bundles, the monodromy group of the Hitchin fibration is generated by Picard–Lefschetz transformations associated with vanishing cycles of singular spectral curves. The basic transvection is

$d$5

Here monodromy graphs encode the intersection pattern of a distinguished basis of vanishing cycles. Their vertices correspond to vanishing cycles, and edges encode intersection numbers, typically distinguishing the vanishing-lattice type. This allows the $d$6 monodromy group to be identified as a skew-symmetric vanishing lattice in the sense of Janssen, with explicit types $d$7, $d$8, $d$9, and $G$0 depending on $G$1 and $G$2. An application is that the monodromy-invariant part of the cohomology of a regular Hitchin fibre is generated by powers of the polarization class (Baraglia, 2016).

These moduli-theoretic examples clarify a second widespread feature of monodromy graphs: the graph often does not merely record permutations of a finite set, but instead organises atlases, degeneration patterns, or intersection forms. The graph is then best regarded as a combinatorial skeleton for a monodromy representation rather than as a literal graph of solutions.

5. Dual graphs, resolution graphs, and surface-singularity models

For a proper log curve $G$3 over the standard log point, the dual graph has vertices $G$4 indexed by irreducible components and edges $G$5 indexed by double points. The log de Rham cohomology $G$6 admits a combinatorial monodromy operator defined on a class represented by $G$7 by

$G$8

The residue condition

$G$9

holds for every node, and the invariant part is identified with Du Bois cohomology through an exact sequence that is an algebraic analogue of the local invariant cycles sequence. In the complex semistable case, this construction recovers the classical monodromy operator $F_{p_v}$0 (Gatti, 2018).

Log geometry generalises this perspective. For a log smooth degeneration over the standard log disc, the Kato–Nakayama construction attaches a topological space $F_{p_v}$1 to the log scheme and allows one to recover the topology of the germ of the family from the log special fiber alone. In the case of curves, the data are described as essentially equivalent to those encoded by the dual graph of a semistable degeneration, including the monodromy pairing and the Picard–Lefschetz formula. The nearby-cycle spectral sequence and its $F_{p_v}$2-differentials admit combinatorial descriptions in terms of the log structures, and variants exist in the Kummer étale setting (Achinger et al., 2018).

Resolution graphs supply another monodromy-graph formalism. For the space monomial curves considered as Cartier divisors on generic embedding surfaces, an embedded $F_{p_v}$3-resolution is constructed by successive weighted blow-ups. The exceptional divisors form a tree of rational curves, and a generalised A’Campo formula computes the zeta function of monodromy from the strata $F_{p_v}$4: $F_{p_v}$5 For these curves the monodromy conjecture is proved by analysing the eigenvalues arising from the explicit zeta function (Martín-Morales et al., 2019).

A topological realisation of surface monodromy is provided by tête-à-tête graphs and mixed tête-à-tête graphs. A pure tête-à-tête graph is a metric ribbon graph whose safe walks encode a finite-order mapping class. Mixed tête-à-tête graphs are filtered relative metric ribbon graphs $F_{p_v}$6 with level-dependent safe-walk data, and they realise pseudo-periodic automorphisms with negative screw numbers and positive fractional Dehn twist coefficients at boundaries. The realisation theorem states that such a pseudo-periodic automorphism is induced by a relative mixed tête-à-tête graph. More strongly, mixed tête-à-tête twists are characterised as the monodromies associated with reduced function germs on isolated complex surface singularities, and mixed tête-à-tête twists contain monodromies of irreducible plane curve singularities (Cuadrado et al., 2017, A'Campo et al., 2017).

6. Combinatorial and graph-theoretic variants

Several further notions use “monodromy graph” for explicitly combinatorial objects. For irreducible nodal rational curves on a complete toric surface, the monodromy map

$F_{p_v}$7

acts on the finite set of nodes of a reference curve. Via the order map for rational simple Harnack curves, the nodes are identified with $F_{p_v}$8, and an obstruction map

$F_{p_v}$9

restricts which permutations can occur. The image of monodromy is always a subgroup of the deck transformations $F_{p_i}$00, and for sufficiently ample $F_{p_i}$01 it is exactly that group. The paper also exhibits families with small $F_{p_i}$02 for which the image is strictly smaller than expected (Lang, 2019).

For direct sums $F_{p_i}$03, especially $F_{p_i}$04, the relevant monodromy graph is the Dynkin diagram of vanishing cycles. Its vertices are the join cycles $F_{p_i}$05, arranged in a two-dimensional grid, and its edges are weighted by the Picard–Lefschetz intersection formula. Horizontal symmetry occurs exactly when $F_{p_i}$06 is decomposable, and in the quartic case a further vertical symmetry arises from the map $F_{p_i}$07. The monodromy orbit of a vanishing cycle generates the entire homology unless prevented by these symmetries; for direct sums of fourth-degree polynomials the possibilities are classified graphically and algebraically (Garcia, 2020).

In embedded graph theory, $F_{p_i}$08-monodromy is a permutation attached to a face of a graph embedded in a closed surface, under the assumption that both the graph and its dual are simple. If $F_{p_i}$09 is $F_{p_i}$10-gonal, the $F_{p_i}$11-monodromy $F_{p_i}$12 acts on the oriented edges $F_{p_i}$13. The realisable permutations are characterised exactly by two conditions: $F_{p_i}$14 Thus every permutation of $F_{p_i}$15 satisfying reversibility and no orientation reversal is realised as the $F_{p_i}$16-monodromy of some $F_{p_i}$17-gonal face in a connected simple finite graph embedded in a surface with simple dual (Tyc, 2023).

A more group-theoretic variant studies monodromy groups through Cayley graphs of finite quotients. For a finitely generated monodromy group $F_{p_i}$18 with generating set $F_{p_i}$19, the Cayley graphs $F_{p_i}$20 of congruence quotients $F_{p_i}$21 encode strong and superstrong approximation. Superstrong approximation is the expander condition $F_{p_i}$22 for all large $F_{p_i}$23, and these monodromy graphs transmit group-theoretic expansion to arithmetic consequences such as gonality growth in towers of covers (Ellenberg, 2012).

Taken together, these constructions show that monodromy graphs range from computational data structures to dual graphs, Dynkin diagrams, exchange graphs, dessins, ribbon graphs, and Cayley graphs. The unifying principle is not a fixed definition but the use of finite combinatorial objects to encode how monodromy acts, degenerates, or can be computed.